The document provides an overview of concepts in functional analysis that will be covered in a math camp, including: function spaces, metric spaces, dense subsets, linear spaces, linear functionals, norms, Euclidean spaces, orthogonality, separable spaces, complete metric spaces, Hilbert spaces, and convex functions. Examples are given for each concept to illustrate the definitions.
Math 511 Problem Set 4, due September 21Note Problems 1 tAbramMartino96
Math 511 Problem Set 4, due September 21
Note: Problems 1 through 7 are the ones to be turned in. The remainder of the problems are
for extra functional analytic goodness.
1. Fix a,b ∈ R with a < b. Show that {1, t, t2, . . . , tn} is a linearly independent subset of
C[a,b]. From this conclude that {1, t, t2, t3, . . .} is a linearly independent set in C[a,b]. Give
an example of a function f ∈ C[a,b] so that f /∈ span{1, t, t2, . . .}.
2. Prove that if 1 ≤ p1 ≤ p2 ≤∞ then lp1 ⊆ lp2 .
3. Consider C[0, 2] with the function ‖ ·‖1 defined by
‖f‖1 =
∫ 2
0
|f(x)|dx, for f ∈ C[0, 2].
(a) Prove that ‖ ·‖1 is a norm.
(b) Prove that the normed linear space (C[0, 2],‖·‖1) is not complete (and thus not a Banach
space) by considering the sequence of functions
fn(x) =
1, x ≤ 1 − 1
n
n−nx, 1 − 1
n
< x < 1 + 1
n
−1, x ≥ 1 + 1
n
.
Show these are continuous functions, this sequence is a Cauchy sequence in the metric
derived from ‖ ·‖1, but that this sequence does not converge in C[0, 2] with this metric.
4. Let V be a vector space over R or C. A subset A ⊆ V is convex if for any v,w ∈ A and any
λ ∈ [0, 1] then λv + (1 −λ)w ∈ A, i.e. the segement connecting v and w is also in A.
(a) Let W be a vector subspace of V . Show that W is convex.
(b) Let X be a normed linear space. Show that the unit ball B1(0) is convex.
5. show that c ⊆ l∞ is a vector subspace of l∞ (see 1.5-3 for the definition of c) and so is c0, the
set of all sequences (xn) so that limn→∞ xn = 0.
6. Let 1 ≤ p < ∞ and en ∈ lp be the sequence with 1 in the nth place and 0 in all othe coordinates.
Show that {en : n ∈ N} is a Schauder basis for lp.
7. Now if X is a Banach space and (yn) a sequence in X, prove that
∑∞
n=1 ‖yn‖ < ∞ does imply
the convergence of
∑∞
n=1 yn. Thus in Banach spaces, absolute convergence implies convergence
of the series.
The following questions are for you to think about and not to be turned in.
1001. What is the completion of (0, 1) as a metric subspace of R with the euclidean metric?
Explain.
1002. Show that the discrete metric on a nontrivial vector space cannot be obtained from a norm.
1003. Show that if a normed vector space has a Schauder basis, then the space is separable. (You
can use a similar argument to your proof that lp is separable for 1 ≤ p < ∞.)
1004. Prove the general Hölder inequality: Suppose 1 ≤ r < p < ∞, and assume that
1
p
+
1
q
=
1
r
.
Show that for x = (x1,x2, . . .) and y = (y1,y2, . . .), and if we define the componentwise product
xy = (x1y1,x2y2, . . .), then
‖xy‖r ≤‖x‖p‖y‖q.
You may assume that x ∈ lp and y ∈ lq, although this is not necessary. (Hint: 1 = 1p
r
+ 1q
r
, and
use the regular Hölder inequality on particular sequences).
(Note: We can extend this to let p = r, and in this case q = ∞. The result will still hold.)
1005. Give an example of a subspace of l∞ which is not closed. Repeat for l2. (Hint: Look at
problem 3, p. 70)
1006. Let X be a normed vector space. Show that the convergenc ...
ABSTRACT : In this paper we have studied the behavior of the different limits in some of the dual spaces of function spaces. A good number of results have been established. We have also observed that in few cases we have to take the of a set suitable conditions. We also used the notion of a perfect function space in establishing a few results .As a matter of facts the notions of which are used in this paper are parametric convergent ,parametric limit, projective convergent , projective limit, dual space of a function space and perfect function space in addition of the suitably defined and constructed some of the function spaces.
On Spaces of Entire Functions Having Slow Growth Represented By Dirichlet SeriesIOSR Journals
In this paper spaces of entire function represented by Dirichlet Series have been considered. A
norm has been introduced and a metric has been defined. Properties of this space and a characterization of
continuous linear functionals have been established.
This document discusses signal-space analysis and representation of bandpass signals. It can be summarized as follows:
1) A bandpass real signal x(t) can be represented using its complex envelope x(t) and carrier frequency fc. This results in an in-phase (I) and quadrature-phase (Q) representation of the signal.
2) Signals can be viewed as vectors in a vector space. Basic algebra concepts like groups, fields, and vector spaces are introduced.
3) Key concepts discussed include orthonormal bases, projection theorems, Gram-Schmidt orthonormalization, and representing signals in inner product spaces which allows defining notions of length and angle between signals.
This document discusses convexity properties of the gamma function. It begins with an introduction to the gamma function and its history. It then revisits earlier work by Krull on functional equations of the form f(x+1)-f(x)=g(x). Several lemmas are presented that characterize properties of convex functions satisfying certain conditions. These results are then applied to derive classical and new representations and characterizations of the gamma function based on its convexity.
This document provides the contents page for a book on integral equations. It lists 22 chapters covering topics such as Volterra integral equations, Fredholm integral equations, approximate solution methods, Green's functions, and singular integral equations. It was translated from the Russian original published in 1971 in Moscow by MIR Publishers.
This document contains the solution to a problem involving a sequence of continuously differentiable functions defined by a recurrence relation. The solution shows that:
1) The sequence is monotonically increasing and bounded, so it converges pointwise to a limit function g(x).
2) The limit function g(x) is the unique fixed point of the operator defining the recurrence, and is equal to 1/(1-x).
3) Uniform convergence on compact subsets is proved using Dini's theorem and properties of the operator.
This document provides an introduction to functions and limits. It defines key concepts such as domain, range, and different types of functions including algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate how to find the domain and range of functions, evaluate functions, and draw graphs of functions. Function notation and the concept of a function as a rule that assigns each input to a single output are also explained.
Math 511 Problem Set 4, due September 21Note Problems 1 tAbramMartino96
Math 511 Problem Set 4, due September 21
Note: Problems 1 through 7 are the ones to be turned in. The remainder of the problems are
for extra functional analytic goodness.
1. Fix a,b ∈ R with a < b. Show that {1, t, t2, . . . , tn} is a linearly independent subset of
C[a,b]. From this conclude that {1, t, t2, t3, . . .} is a linearly independent set in C[a,b]. Give
an example of a function f ∈ C[a,b] so that f /∈ span{1, t, t2, . . .}.
2. Prove that if 1 ≤ p1 ≤ p2 ≤∞ then lp1 ⊆ lp2 .
3. Consider C[0, 2] with the function ‖ ·‖1 defined by
‖f‖1 =
∫ 2
0
|f(x)|dx, for f ∈ C[0, 2].
(a) Prove that ‖ ·‖1 is a norm.
(b) Prove that the normed linear space (C[0, 2],‖·‖1) is not complete (and thus not a Banach
space) by considering the sequence of functions
fn(x) =
1, x ≤ 1 − 1
n
n−nx, 1 − 1
n
< x < 1 + 1
n
−1, x ≥ 1 + 1
n
.
Show these are continuous functions, this sequence is a Cauchy sequence in the metric
derived from ‖ ·‖1, but that this sequence does not converge in C[0, 2] with this metric.
4. Let V be a vector space over R or C. A subset A ⊆ V is convex if for any v,w ∈ A and any
λ ∈ [0, 1] then λv + (1 −λ)w ∈ A, i.e. the segement connecting v and w is also in A.
(a) Let W be a vector subspace of V . Show that W is convex.
(b) Let X be a normed linear space. Show that the unit ball B1(0) is convex.
5. show that c ⊆ l∞ is a vector subspace of l∞ (see 1.5-3 for the definition of c) and so is c0, the
set of all sequences (xn) so that limn→∞ xn = 0.
6. Let 1 ≤ p < ∞ and en ∈ lp be the sequence with 1 in the nth place and 0 in all othe coordinates.
Show that {en : n ∈ N} is a Schauder basis for lp.
7. Now if X is a Banach space and (yn) a sequence in X, prove that
∑∞
n=1 ‖yn‖ < ∞ does imply
the convergence of
∑∞
n=1 yn. Thus in Banach spaces, absolute convergence implies convergence
of the series.
The following questions are for you to think about and not to be turned in.
1001. What is the completion of (0, 1) as a metric subspace of R with the euclidean metric?
Explain.
1002. Show that the discrete metric on a nontrivial vector space cannot be obtained from a norm.
1003. Show that if a normed vector space has a Schauder basis, then the space is separable. (You
can use a similar argument to your proof that lp is separable for 1 ≤ p < ∞.)
1004. Prove the general Hölder inequality: Suppose 1 ≤ r < p < ∞, and assume that
1
p
+
1
q
=
1
r
.
Show that for x = (x1,x2, . . .) and y = (y1,y2, . . .), and if we define the componentwise product
xy = (x1y1,x2y2, . . .), then
‖xy‖r ≤‖x‖p‖y‖q.
You may assume that x ∈ lp and y ∈ lq, although this is not necessary. (Hint: 1 = 1p
r
+ 1q
r
, and
use the regular Hölder inequality on particular sequences).
(Note: We can extend this to let p = r, and in this case q = ∞. The result will still hold.)
1005. Give an example of a subspace of l∞ which is not closed. Repeat for l2. (Hint: Look at
problem 3, p. 70)
1006. Let X be a normed vector space. Show that the convergenc ...
ABSTRACT : In this paper we have studied the behavior of the different limits in some of the dual spaces of function spaces. A good number of results have been established. We have also observed that in few cases we have to take the of a set suitable conditions. We also used the notion of a perfect function space in establishing a few results .As a matter of facts the notions of which are used in this paper are parametric convergent ,parametric limit, projective convergent , projective limit, dual space of a function space and perfect function space in addition of the suitably defined and constructed some of the function spaces.
On Spaces of Entire Functions Having Slow Growth Represented By Dirichlet SeriesIOSR Journals
In this paper spaces of entire function represented by Dirichlet Series have been considered. A
norm has been introduced and a metric has been defined. Properties of this space and a characterization of
continuous linear functionals have been established.
This document discusses signal-space analysis and representation of bandpass signals. It can be summarized as follows:
1) A bandpass real signal x(t) can be represented using its complex envelope x(t) and carrier frequency fc. This results in an in-phase (I) and quadrature-phase (Q) representation of the signal.
2) Signals can be viewed as vectors in a vector space. Basic algebra concepts like groups, fields, and vector spaces are introduced.
3) Key concepts discussed include orthonormal bases, projection theorems, Gram-Schmidt orthonormalization, and representing signals in inner product spaces which allows defining notions of length and angle between signals.
This document discusses convexity properties of the gamma function. It begins with an introduction to the gamma function and its history. It then revisits earlier work by Krull on functional equations of the form f(x+1)-f(x)=g(x). Several lemmas are presented that characterize properties of convex functions satisfying certain conditions. These results are then applied to derive classical and new representations and characterizations of the gamma function based on its convexity.
This document provides the contents page for a book on integral equations. It lists 22 chapters covering topics such as Volterra integral equations, Fredholm integral equations, approximate solution methods, Green's functions, and singular integral equations. It was translated from the Russian original published in 1971 in Moscow by MIR Publishers.
This document contains the solution to a problem involving a sequence of continuously differentiable functions defined by a recurrence relation. The solution shows that:
1) The sequence is monotonically increasing and bounded, so it converges pointwise to a limit function g(x).
2) The limit function g(x) is the unique fixed point of the operator defining the recurrence, and is equal to 1/(1-x).
3) Uniform convergence on compact subsets is proved using Dini's theorem and properties of the operator.
This document provides an introduction to functions and limits. It defines key concepts such as domain, range, and different types of functions including algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate how to find the domain and range of functions, evaluate functions, and draw graphs of functions. Function notation and the concept of a function as a rule that assigns each input to a single output are also explained.
This document provides an introduction to functions and their key concepts. It defines what a function is, using examples to illustrate functions that relate variables. Functions have a domain and range, and can be represented graphically. Common types of functions are discussed, including algebraic functions like polynomials and rational functions, as well as trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Methods for determining a function's domain and range and drawing its graph are presented.
This document provides an introduction to functions and their key concepts. It defines a function as a rule that assigns each element in one set to a unique element in another set. Functions can be represented graphically and algebraically. Common types of functions discussed include polynomial, linear, constant, rational, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate domain, range, and graphing of different function types.
This document contains exercises related to dynamical systems and periodic points. It includes the following summaries:
1. The doubling map on the circle has 2n-1 periodic points of period n. Its periodic points are dense.
2. The map f(x)=|x-2| has a fixed point at x=1. Other periodic and pre-periodic points are [0,2]\{1\} of period 2 and (-∞,0)∪(2,+∞) which are pre-periodic.
3. Expanding maps of the circle are topologically mixing since intervals get longer under iteration, eventually covering the entire circle.
This document discusses Fourier series and their applications. It contains the following key points:
1. Fourier introduced Fourier series to solve heat equations through metal plates, expressing functions as infinite sums of sines and cosines.
2. Sine and cosine functions are orthogonal and periodic, allowing any piecewise continuous periodic function to be represented by a Fourier series.
3. The Euler-Fourier formulas relate the Fourier coefficients to the function, allowing the coefficients to be determined.
4. Even functions only have cosine terms, odd only sine, and the Fourier series converges to the average at discontinuities for piecewise continuous functions.
This document outlines the contents of a Mathematics II course, including five units: vector calculus, Fourier series and Fourier transforms, interpolation and curve fitting, solutions to algebraic/transcendental equations and linear systems of equations, and numerical integration and solutions to differential equations. It lists three textbooks and four references used in the course. It then provides examples and explanations of key concepts from the first two units, including vector differential operators, gradient, divergence, curl, and Fourier series representations of functions.
This document provides definitions and examples of various types of numbers and functions. It discusses:
- Number sets including natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
- Types of intervals such as closed, open, and semi-open/semi-closed.
- Definitions of a function, including domain, co-domain, and range. Methods of representing functions include mapping, algebraic, and ordered pairs.
- Classification of functions as algebraic vs. transcendental, even vs. odd, explicit vs. implicit, continuous vs. discontinuous, and increasing vs. decreasing.
- Properties of even and odd functions are also discussed.
1. Prove that the function f(x) = x^2 if x is rational, 0 otherwise, is differentiable at 0 but discontinuous everywhere else.
2. Use the Chain Rule to find an expression for the derivative of the inverse function g(f(c)) in terms of f'(c), given that f and g are inverse bijective functions between intervals with f differentiable at c and f(c) = 0, and g differentiable at f(c).
3. Prove several rules for derivatives, including the Power Rule and derivatives of composite functions.
Measure Theory and important points with bookletNaeemAhmad289736
The document provides a summary of key concepts in measure theory and Lebesgue integration. In 3 sentences: It defines topologies and σ-algebras, and describes measurable functions and spaces. It introduces measures, including positive measures and the Lebesgue measure. It covers integrals of simple and Lebesgue integrable functions, and theorems like monotone convergence, dominated convergence, and Fubini's theorem.
This document discusses different types of relations and functions. It defines equivalence relations, identity relations, empty relations, universal relations, one-to-one functions, onto functions, bijective functions, composition of functions, and invertible functions. It provides examples to illustrate these concepts.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
This document summarizes a research article that reviews integration in Banach spaces, with a focus on the Bochner integral, generalized derivatives, and generalized gradients. It presents key definitions related to strongly measurable functions, the Bochner integral, Lp spaces of functions from an interval to a Banach space, generalized derivatives, and monotone operators. It also states theorems regarding Banach and Hilbert space properties, generalized derivative properties under weak convergence, and finite extensions of the Bochner integral for sums and products. The main results develop these finite extensions of the Bochner integral.
This document summarizes and reviews concepts related to Frechet derivatives. It begins by defining Frechet derivatives on Banach spaces and their properties such as differentiability of compositions of functions. It then discusses applications to ordinary differential equations, including the inverse function theorem. Higher order Frechet derivatives and their properties are also introduced. The document concludes by stating results on the mean value theorem, Taylor's theorem, and Riemannian integration as they apply to Frechet derivatives.
This document introduces Fourier series and their motivation. Joseph Fourier realized that many physical phenomena, like heat diffusion, could be modeled using partial differential equations. He developed a method of separation of variables to solve these equations, which led him to represent functions as infinite sums of sines and cosines, now known as Fourier series. The document outlines Fourier's approach, showing how assuming a solution of the form of separated variables leads to an eigenvalue problem. The eigenfunctions form a basis to represent more general functions as Fourier series. The Fourier coefficients that define a particular function can be determined by integrating the function against the eigenfunctions.
On Frechet Derivatives with Application to the Inverse Function Theorem of Or...BRNSS Publication Hub
This document summarizes and reviews concepts related to Frechet derivatives. It begins by defining Frechet derivatives on Banach spaces and their properties such as differentiability of compositions of functions. It then discusses applications to ordinary differential equations, including the inverse function theorem. Higher order Frechet derivatives and partial derivatives on product spaces are also introduced. The document concludes by discussing concepts like the mean value theorem and Taylor's theorem in the context of Frechet derivatives.
The document discusses domain and range of functions. It provides examples of determining the domain and range from graphs of functions and from algebraic rules that define functions. The domain of a function is the set of permissible input values, while the range is the set of permissible output values. Examples show how to identify the domain and range from graphs by considering limits, and how to determine them algebraically by manipulating equations to solve for variables.
This document presents theorems and results regarding hypercyclic operators on the space Hbc(E), where E is a Banach space. Theorem 3.1 shows that the collection of functions {eφ : φ ∈ E*} forms an independently linear subset of Hbc(E). Theorem 3.2 proves that the span of {eφ : φ ∈ U} is dense in Hbc(E), where U is an open subset of E*. Theorem 3.3 demonstrates that if φ is an entire function of exponential type, then the operator φα(D) is hypercyclic on Hbc(E). The document also provides two corollaries: if E has a separable dual, then
This document provides an overview of preliminary topological concepts needed for applied mathematics. It defines topological spaces and metric spaces, and introduces key topological notions like open and closed sets, bases for topologies, convergence of sequences, accumulation points, interior and closure of sets, and dense sets. Metric spaces are shown to induce a natural topological structure, though not all topologies come from a metric. Examples are provided to illustrate various definitions and properties.
Math 1102-ch-3-lecture note Fourier Series.pdfhabtamu292245
1. The document discusses Fourier series and orthogonal functions. It defines orthogonal functions and provides examples of orthogonal function sets, such as cosine and sine functions.
2. The chief advantage of orthogonal functions is that they allow functions to be represented as generalized Fourier series expansions. The orthogonality of the functions helps determine the Fourier coefficients in a simple way using integrals.
3. Euler's formulae give the expressions for calculating the Fourier coefficients a0, an, and bn of a periodic function f(x) from its values over one period using integrals of f(x) multiplied by cosine and sine terms.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
This document provides an introduction to functions and their key concepts. It defines what a function is, using examples to illustrate functions that relate variables. Functions have a domain and range, and can be represented graphically. Common types of functions are discussed, including algebraic functions like polynomials and rational functions, as well as trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Methods for determining a function's domain and range and drawing its graph are presented.
This document provides an introduction to functions and their key concepts. It defines a function as a rule that assigns each element in one set to a unique element in another set. Functions can be represented graphically and algebraically. Common types of functions discussed include polynomial, linear, constant, rational, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate domain, range, and graphing of different function types.
This document contains exercises related to dynamical systems and periodic points. It includes the following summaries:
1. The doubling map on the circle has 2n-1 periodic points of period n. Its periodic points are dense.
2. The map f(x)=|x-2| has a fixed point at x=1. Other periodic and pre-periodic points are [0,2]\{1\} of period 2 and (-∞,0)∪(2,+∞) which are pre-periodic.
3. Expanding maps of the circle are topologically mixing since intervals get longer under iteration, eventually covering the entire circle.
This document discusses Fourier series and their applications. It contains the following key points:
1. Fourier introduced Fourier series to solve heat equations through metal plates, expressing functions as infinite sums of sines and cosines.
2. Sine and cosine functions are orthogonal and periodic, allowing any piecewise continuous periodic function to be represented by a Fourier series.
3. The Euler-Fourier formulas relate the Fourier coefficients to the function, allowing the coefficients to be determined.
4. Even functions only have cosine terms, odd only sine, and the Fourier series converges to the average at discontinuities for piecewise continuous functions.
This document outlines the contents of a Mathematics II course, including five units: vector calculus, Fourier series and Fourier transforms, interpolation and curve fitting, solutions to algebraic/transcendental equations and linear systems of equations, and numerical integration and solutions to differential equations. It lists three textbooks and four references used in the course. It then provides examples and explanations of key concepts from the first two units, including vector differential operators, gradient, divergence, curl, and Fourier series representations of functions.
This document provides definitions and examples of various types of numbers and functions. It discusses:
- Number sets including natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
- Types of intervals such as closed, open, and semi-open/semi-closed.
- Definitions of a function, including domain, co-domain, and range. Methods of representing functions include mapping, algebraic, and ordered pairs.
- Classification of functions as algebraic vs. transcendental, even vs. odd, explicit vs. implicit, continuous vs. discontinuous, and increasing vs. decreasing.
- Properties of even and odd functions are also discussed.
1. Prove that the function f(x) = x^2 if x is rational, 0 otherwise, is differentiable at 0 but discontinuous everywhere else.
2. Use the Chain Rule to find an expression for the derivative of the inverse function g(f(c)) in terms of f'(c), given that f and g are inverse bijective functions between intervals with f differentiable at c and f(c) = 0, and g differentiable at f(c).
3. Prove several rules for derivatives, including the Power Rule and derivatives of composite functions.
Measure Theory and important points with bookletNaeemAhmad289736
The document provides a summary of key concepts in measure theory and Lebesgue integration. In 3 sentences: It defines topologies and σ-algebras, and describes measurable functions and spaces. It introduces measures, including positive measures and the Lebesgue measure. It covers integrals of simple and Lebesgue integrable functions, and theorems like monotone convergence, dominated convergence, and Fubini's theorem.
This document discusses different types of relations and functions. It defines equivalence relations, identity relations, empty relations, universal relations, one-to-one functions, onto functions, bijective functions, composition of functions, and invertible functions. It provides examples to illustrate these concepts.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
This document summarizes a research article that reviews integration in Banach spaces, with a focus on the Bochner integral, generalized derivatives, and generalized gradients. It presents key definitions related to strongly measurable functions, the Bochner integral, Lp spaces of functions from an interval to a Banach space, generalized derivatives, and monotone operators. It also states theorems regarding Banach and Hilbert space properties, generalized derivative properties under weak convergence, and finite extensions of the Bochner integral for sums and products. The main results develop these finite extensions of the Bochner integral.
This document summarizes and reviews concepts related to Frechet derivatives. It begins by defining Frechet derivatives on Banach spaces and their properties such as differentiability of compositions of functions. It then discusses applications to ordinary differential equations, including the inverse function theorem. Higher order Frechet derivatives and their properties are also introduced. The document concludes by stating results on the mean value theorem, Taylor's theorem, and Riemannian integration as they apply to Frechet derivatives.
This document introduces Fourier series and their motivation. Joseph Fourier realized that many physical phenomena, like heat diffusion, could be modeled using partial differential equations. He developed a method of separation of variables to solve these equations, which led him to represent functions as infinite sums of sines and cosines, now known as Fourier series. The document outlines Fourier's approach, showing how assuming a solution of the form of separated variables leads to an eigenvalue problem. The eigenfunctions form a basis to represent more general functions as Fourier series. The Fourier coefficients that define a particular function can be determined by integrating the function against the eigenfunctions.
On Frechet Derivatives with Application to the Inverse Function Theorem of Or...BRNSS Publication Hub
This document summarizes and reviews concepts related to Frechet derivatives. It begins by defining Frechet derivatives on Banach spaces and their properties such as differentiability of compositions of functions. It then discusses applications to ordinary differential equations, including the inverse function theorem. Higher order Frechet derivatives and partial derivatives on product spaces are also introduced. The document concludes by discussing concepts like the mean value theorem and Taylor's theorem in the context of Frechet derivatives.
The document discusses domain and range of functions. It provides examples of determining the domain and range from graphs of functions and from algebraic rules that define functions. The domain of a function is the set of permissible input values, while the range is the set of permissible output values. Examples show how to identify the domain and range from graphs by considering limits, and how to determine them algebraically by manipulating equations to solve for variables.
This document presents theorems and results regarding hypercyclic operators on the space Hbc(E), where E is a Banach space. Theorem 3.1 shows that the collection of functions {eφ : φ ∈ E*} forms an independently linear subset of Hbc(E). Theorem 3.2 proves that the span of {eφ : φ ∈ U} is dense in Hbc(E), where U is an open subset of E*. Theorem 3.3 demonstrates that if φ is an entire function of exponential type, then the operator φα(D) is hypercyclic on Hbc(E). The document also provides two corollaries: if E has a separable dual, then
This document provides an overview of preliminary topological concepts needed for applied mathematics. It defines topological spaces and metric spaces, and introduces key topological notions like open and closed sets, bases for topologies, convergence of sequences, accumulation points, interior and closure of sets, and dense sets. Metric spaces are shown to induce a natural topological structure, though not all topologies come from a metric. Examples are provided to illustrate various definitions and properties.
Math 1102-ch-3-lecture note Fourier Series.pdfhabtamu292245
1. The document discusses Fourier series and orthogonal functions. It defines orthogonal functions and provides examples of orthogonal function sets, such as cosine and sine functions.
2. The chief advantage of orthogonal functions is that they allow functions to be represented as generalized Fourier series expansions. The orthogonality of the functions helps determine the Fourier coefficients in a simple way using integrals.
3. Euler's formulae give the expressions for calculating the Fourier coefficients a0, an, and bn of a periodic function f(x) from its values over one period using integrals of f(x) multiplied by cosine and sine terms.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
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তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
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A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
2. About the primer
Goal To briefly review concepts in functional analysis that
will be used throughout the course.∗ The following
concepts will be described
1. Function spaces
2. Metric spaces
3. Dense subsets
4. Linear spaces
5. Linear functionals
∗The definitions and concepts come primarily from “Introductory Real
Analysis” by Kolmogorov and Fomin (highly recommended).
3. 6. Norms and semi-norms of linear spaces
7. Euclidean spaces
8. Orthogonality and bases
9. Separable spaces
10. Complete metric spaces
11. Hilbert spaces
12. Riesz representation theorem
13. Convex functions
14. Lagrange multipliers
4. Function space
A function space is a space made of functions. Each
function in the space can be thought of as a point. Ex-
amples:
1. C[a, b], the set of all real-valued continuous functions
in the interval [a, b];
2. L1[a, b], the set of all real-valued functions whose ab-
solute value is integrable in the interval [a, b];
3. L2[a, b], the set of all real-valued functions square inte-
grable in the interval [a, b]
Note that the functions in 2 and 3 are not necessarily
continuous!
5. Metric space
By a metric space is meant a pair (X, ρ) consisting of a
space X and a distance ρ, a single-valued, nonnegative,
real function ρ(x, y) defined for all x, y ∈ X which has the
following three properties:
1. ρ(x, y) = 0 iff x = y;
2. ρ(x, y) = ρ(y, x);
3. Triangle inequality: ρ(x, z) ≤ ρ(x, y) + ρ(y, z)
6. Examples
1. The set of all real numbers with distance
ρ(x, y) = |x − y|
is the metric space IR1.
2. The set of all ordered n-tuples
x = (x1, ..., xn)
of real numbers with distance
ρ(x, y) =
v
u
u
t
n
X
i=1
(xi − yi)2
is the metric space IRn.
7. 3. The set of all functions satisfying the criteria
Z
f2(x)dx < ∞
with distance
ρ(f1(x), f2(x)) =
sZ
(f1(x) − f2(x))2dx
is the metric space L2(IR).
4. The set of all probability densities with Kullback-Leibler
divergence
ρ(p1(x), p2(x)) =
Z
ln
p1(x)
p2(x)
p1(x)dx
is not a metric space. The divergence is not symmetric
ρ(p1(x), p2(x)) 6= ρ(p2(x), p1(x)).
8. Dense
A point x ∈ IR is called a contact point of a set A ∈ IR if
every ball centered at x contains at least one point of A.
The set of all contact points of a set A denoted by Ā is
called the closure of A.
Let A and B be subspaces of a metric space IR. A is said
to be dense in B if B ⊂ Ā. In particular A is said to be
everywhere dense in IR if Ā = R.
9. Examples
1. The set of all rational points is dense in the real line.
2. The set of all polynomials with rational coefficients is
dense in C[a, b].
3. The RKHS induced by the gaussian kernel on [a, b] in
dense in L2[a, b]
Note: A hypothesis space that is dense in L2 is a desired
property of any approximation scheme.
10. Linear space
A set L of elements x, y, z, ... is a linear space if the fol-
lowing three axioms are satisfied:
1. Any two elements x, y ∈ L uniquely determine a third
element in x + y ∈ L called the sum of x and y such
that
(a) x + y = y + x (commutativity)
(b) (x + y) + z = x + (y + z) (associativity)
(c) An element 0 ∈ L exists for which x + 0 = x for all
x ∈ L
(d) For every x ∈ L there exists an element −x ∈ L
with the property x + (−x) = 0
11. 2. Any number α and any element x ∈ L uniquely deter-
mine an element αx ∈ L called the product such that
(a) α(βx) = β(αx)
(b) 1x = x
3. Addition and multiplication follow two distributive laws
(a)(α + β)x = αx + βx
(b)α(x + y) = αx + αy
12. Linear functional
A functional, F, is a function that maps another function
to a real-value
F : f → IR.
A linear functional defined on a linear space L, satisfies the
following two properties
1. Additive: F(f + g) = F(f) + F(g) for all f, g ∈ L
2. Homogeneous: F(αf) = αF(f)
13. Examples
1. Let IRn be a real n-space with elements x = (x1, ..., xn),
and a = (a1, ..., an) be a fixed element in IRn. Then
F(x) =
n
X
i=1
aixi
is a linear functional
2. The integral
F[f(x)] =
Z b
a
f(x)p(x)dx
is a linear functional
3. Evaluation functional: another linear functional is the
14. Dirac delta function
δt[f(·)] = f(t).
Which can be written
δt[f(·)] =
Z b
a
f(x)δ(x − t)dx.
4. Evaluation functional: a positive definite kernel in a
RKHS
Ft[f(·)] = (Kt, f) = f(t).
This is simply the reproducing property of the RKHS.
15. Normed space
A normed space is a linear (vector) space N in which a
norm is defined. A nonnegative function k · k is a norm iff
∀f, g ∈ N and α ∈ IR
1. kfk ≥ 0 and kfk = 0 iff f = 0;
2. kf + gk ≤ kfk + kgk;
3. kαfk = |α| kfk.
Note, if all conditions are satisfied except kfk = 0 iff f = 0
then the space has a seminorm instead of a norm.
16. Measuring distances in a normed space
In a normed space N, the distance ρ between f and g, or
a metric, can be defined as
ρ(f, g) = kg − fk.
Note that ∀f, g, h ∈ N
1. ρ(f, g) = 0 iff f = g.
2. ρ(f, g) = ρ(g, f).
3. ρ(f, h) ≤ ρ(f, g) + ρ(g, h).
17. Example: continuous functions
A norm in C[a, b] can be established by defining
kfk = max
a≤t≤b
|f(t)|.
The distance between two functions is then measured as
ρ(f, g) = max
a≤t≤b
|g(t) − f(t)|.
With this metric, C[a, b] is denoted as C.
18. Examples (cont.)
A norm in L1[a, b] can be established by defining
kfk =
Z b
a
|f(t)|dt.
The distance between two functions is then measured as
ρ(f, g) =
Z b
a
|g(t) − f(t)|dt.
With this metric, L1[a, b] is denoted as L1.
19. Examples (cont.)
A norm in C2[a, b] and L2[a, b] can be established by defining
kfk =
Z b
a
f2
(t)dt
!1/2
.
The distance between two functions now becomes
ρ(f, g) =
Z b
a
(g(t) − f(t))2
dt
!1/2
.
With this metric, C2[a, b] and L2[a, b] are denoted as C2
and L2 respectively.
20. Euclidean space
A Euclidean space is a linear (vector) space E in which a
dot product is defined. A real valued function (·, ·) is a dot
product iff ∀f, g, h ∈ E and α ∈ IR
1. (f, g) = (g, f);
2. (f + g, h) = (f, h∗
) + (g, h) and (αf, g) = α(f, g);
3. (f, f) ≥ 0 and (f, f) = 0 iff f = 0.
A Euclidean space becomes a normed linear space when
equipped with the norm
kfk =
q
(f, f).
21. Orthogonal systems and bases
A set of nonzero vectors {xα} in a Euclidean space E is
said to be an orthogonal system if
(xα, xβ) = 0 for α 6= β
and an orthonormal system if
(xα, xβ) = 0 for α 6= β
(xα, xβ) = 1 for α = β.
An orthogonal system {xα} is called an orthogonal basis
if it is complete (the smallest closed subspace containing
{xα} is the whole space E). A complete orthonormal sys-
tem is called an orthonormal basis.
22. Examples
1. IRn is a real n-space, the set of n-tuples x = (x1, ..., xn),
y = (y1, ..., yn). If we define the dot product as
(x, y) =
n
X
i=1
xiyi
we get Euclidean n-space. The corresponding norms
and distances in IRn are
kxk =
v
u
u
t
n
X
i=1
x2
i
ρ(x, y) = kx − yk =
v
u
u
t
n
X
i=1
(xi − yi)2.
23. The vectors
e1 = (1, 0, 0, ...., 0)
e2 = (0, 1, 0, ...., 0)
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
en = (0, 0, 0, ...., 1)
form an orthonormal basis in IRn.
2. The space l2 with elements x = (x1, x2, ..., xn, ....), y =
(y1, y2, ..., yn, ....), ..., where
∞
X
i=1
x2
i < ∞,
∞
X
i=1
y2
i < ∞, ..., ...,
becomes an infinite-dimensional Euclidean space when
equipped with the dot product
(x, y) =
∞
X
i=1
xiyi.
24. The simplest orthonormal basis in l2 consists of vectors
e1 = (1, 0, 0, 0, ...)
e2 = (0, 1, 0, 0, ...)
e3 = (0, 0, 1, 0, ...)
e4 = (0, 0, 0, 1, ...)
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
there are an infinite number of these bases.
3. The space C2[a, b] consisting of all continuous functions
on [a, b] equipped with the dot product
(f, g) =
Z b
a
f(t)g(t)dt
is another example of Euclidean space.
25. An important example of orthogonal bases in this space
is the following set of functions
1, cos
2πnt
b − a
, sin
2πnt
b − a
(n = 1, 2, ...).
26. Cauchy-Schwartz inequality
Let H be an Euclidean space. Then ∀f, g ∈ H, it holds
|(f, g)| ≤ kfk kgk
Sketch of the proof. The case f ∝ g is trivial, hence let
us assume the opposite is true. For all x ∈ IR,
0 < (f + xg, f + xg) = x2 kgk2 + 2x (f, g) + kfk2,
since the quadratic polynomial of x above has no zeroes,
the discriminant ∆ must be negative
0 > ∆/4 = (f, g)2 − kfk2 kgk2.
27. Separable
A metric space is said to be separable if it has a countable
everywhere dense subset.
Examples:
1. The spaces IR1, IRn, L2[a, b], and C[a, b] are all separa-
ble.
2. The set of real numbers is separable since the set of
rational numbers is a countable subset of the reals and
the set of rationals is is everywhere dense.
28. Completeness
A sequence of functions fn is fundamental if ∀ 0 ∃N
such that
∀n and m N, ρ(fn, fm) .
A metric space is complete if all fundamental sequences
converge to a point in the space.
C, L1
, and L2
are complete. That C2 is not complete,
instead, can be seen through a counterexample.
29. Incompleteness of C2
Consider the sequence of functions (n = 1, 2, ...)
φn(t) =
−1 if − 1 ≤ t −1/n
nt if − 1/n ≤ t 1/n
1 if 1/n ≤ t ≤ 1
and assume that φn converges to a continuous function φ
in the metric of C2. Let
f(t) =
(
−1 if − 1 ≤ t 0
1 if 0 ≤ t ≤ 1
30. Incompleteness of C2 (cont.)
Clearly,
Z
(f(t) − φ(t))2
dt
1/2
≤
Z
(f(t) − φn(t))2
dt
1/2
+
Z
(φn(t) − φ(t))2
dt
1/2
.
Now the l.h.s. term is strictly positive, because f(t) is not
continuous, while for n → ∞ we have
Z
(f(t) − φn(t))2
dt → 0.
Therefore, contrary to what assumed, φn cannot converge
to φ in the metric of C2.
31. Completion of a metric space
Given a metric space IR with closure ¯
IR, a complete metric
space IR∗ is called a completion of IR if IR ⊂ IR∗ and
¯
IR = IR∗.
Examples
1. The space of real numbers is the completion of the
space of rational numbers.
2. L2 is the completion of the functional space C2.
32. Hilbert space
A Hilbert space is a Euclidean space that is complete,
separable, and generally infinite-dimensional.
A Hilbert space is a set H of elements f, g, ... for which
1. H is a Euclidean space equipped with a scalar product
2. H is complete with respect to metric ρ(f, g) = kf − gk
3. H is separable (contains a countable everywhere dense
subset)
4. (generally) H is infinite-dimensional.
l2 and L2 are examples of Hilbert spaces.
33. Evaluation functionals
A linear evaluation functional is a linear functional Ft that
evaluates each function in the space at the point t, or
Ft[f] = f(t)
Ft[f + g] = f(t) + g(t).
The functional is bounded if there exists a M s.t.
|Ft[f]| = |f(t)| ≤ MkfkHil ∀t
for all f where k · kHil is the norm in the Hilbert space.
34. Evaluation functionals in Hilbert space
The evaluation functional is not bounded in the familiar
Hilbert space L2([0, 1]), no such M exists and in fact ele-
ments of L2([0, 1]) are not even defined pointwise.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
1
2
3
4
5
6
x
f(x)
35. Evaluation functionals in Hilbert space
In the following pictures the two functions have the same
norm but they are very different on sets of zero measure
−10 −8 −6 −4 −2 0 2 4 6 8 10
−2
−1.5
−1
−0.5
0
0.5
1
function 1
x
f(x)
−10 −8 −6 −4 −2 0 2 4 6 8 10
−1
−0.5
0
0.5
1
1.5
2
2.5
x
f(x)
function 2
36. Riesz Representation Theorem
For every bounded linear functional F on a Hilbert space
H, there is a unique v ∈ H such that
F[x] = (x, v)H, ∀x ∈ H
37. Convex sets
A set X ∈ IRn is convex if
∀x1, x2 ∈ X, ∀λ ∈ [0, 1], λx1 + (1 − λ)x2 ∈ X.
A set is convex if, given any two points in the set, the line
segment connecting them lies entirely inside the set.
39. Convex Functions
A function f : IRn → IR is convex if:
For any x1 and x2 in the domain of f, for any λ ∈ [0, 1],
f(λx1 + (1 − λ)x2) ≤ λf(x1) + (1 − λ)f(x2).
A function is strictly convex if we replace “≤” with “”.
43. Why We Like Convex Functions
Unconstrained convex functions (convex functions where
the domain is all of IRn) are easy to minimize. Convex
functions are differentiable almost everywhere. Directional
derivatives always exist. If we cannot improve our solution
by moving locally, we are at the optimum. If we cannot
find a direction that improves our solution, we are at the
optimum.
44. Why We Like Convex Sets
Convex functions over convex sets (a convex domain) are
also easy to minimize. If the set and the functions are both
convex, if we cannot find a direction which we are able to
move in which decreases the function, we are done. Local
optima are global optima.
45. Optimizing a Convex Function Over a
Convex and a Non-Convex Set
f(x,y) = -x + -y
Global Optima
Local Optimum
46. Existence and uniqueness of minimum
Let f : IRn → IR be a strictly convex function.
The function f is said to be coercive if
lim
kxk→+∞
f(x) = +∞.
Strictly convex and coercive functions have exactly one
local (global) minimum.
47. Local condition on the minimum
If the convex function f is differentiable, its gradient ∇f is
null on the minimum x0.
Even if the gradient does not exist, the subgradient ∂f
always exists.
The subgradient of f in x is defined by
∂f(x) = {w ∈ IRn|∀x0 ∈ IRn, f(x0) ≥ f(x) + w · (x0 − x)},
On the minimum x0, it holds
0 ∈ ∂f(x0),
48. Lagrange multiplier’s technique
Lagrange multiplier’s technique allows the reduction of the
constrained minimization problem
Minimize I(x)
subject to Φ(x) ≤ m (for some m)
to the unconstrained minimization problem
Minimize J(x) = I(x) + λΦ(x) (for some λ ≥ 0)
49. Geometric intuition
The fact that ∇I does not vanish in the interior of the
domain implies that the constrained minimum x̄ must lie
on the domain’s boundary (the level curve Φ(x) = m).
Therefore, at the point x̄ the component of ∇I along the
tangent to the curve Φ = m vanishes.
But since the tangent to Φ = m is orthogonal to ∇Φ, we
have that at the point x̄, ∇Φ and ∇I are parallel, or
∇I(x̄) ∝ ∇Φ(x̄).
50. Geometric intuition (Cont)
We thus introduce a parameter λ ≥ 0, called Lagrange
multiplier, and consider the problem of finding the uncon-
strained minimum xλ of
J(x) = I(x) + λΦ(x)
as a function of λ.
By setting ∇J = 0, we actually look for the points where
∇I and ∇Φ are parallel. The idea is to find all such points
and then check which of them lie on the curve Φ = m.