A' Reference book: KREYSZIG B[a,b] Introductory Functional Analysis with Applications Complement of a set A 18, 609 Transpose of a matrix A 113 Space of bounded functions 228 Space of bounded function 11 Space of functions of bounded variation 226 Space of bounded linear operators 118 B(A) BV4, b B(X, Y) B() Open ball 18 Closed bull 18 e A sequence space 34 C C[a, b] став CCX, Y) A sequence spuce 70 Complex plane of the field of complex numbers 6, 51 Unitary-space 6 Space of continuous functions 7 Space of continuously differentiuble functions 110 Space of compact linear operators 411 Domain of an operator 83 Dimension of a space X 54 Kroaecker delta 114 Spectral family 494 Norm of a bounded linear functional 104 Graph of an operator T 292 Identity operator 84 Infimum (greatest lower bound) 619 (T) 2(x,y) Distance from a toy 3 dim X b 2-(E₁) In (T) 1 inf Lab 1 r L(X, Y) M (T) " и A function space 62 A sequence space 11 A sequence space 6 A space of linear operators 118 Annihilator of a set M 148 Null space of an operator T 83 Zero operator 84 Empty set 609 1.3-1 Definition (Ball and sphere). Given a point xoeX and a real number r>0, we define three types of sets: (a) B(xo; ) {xEX | d(x, xo)0 there is a 8>0 such that (see Fig. 6) d(Tx, Txo)<ε for all x satisfying d(x, xo)<8. T is said to be continuous if it is continuous at every point of X. ■ 1.3-4 Theorem (Continuous mapping). A mapping T of a metric space X into a metric space Y is continuous if and only if the inverse image of any open subset of Y is an open subset of X. All the required definitions and theorems are attached, now need solution to given questions, stop copy pasting anything, I want fresh correct solutions, if you do not know the answers Problem 5: Measure-Theoretic Integration on Product Spaces Statement: Let (X, Ex.) and (Y, Ey,) be a-finite measure spaces. Consider the product a-algebra Σ. & Σ. on. Χ Χ Υ and the product measure μ. Χ. ν. Tasks: 1. Fubini's Theorem for Non-Negative Functions: Prove that for any non-negative measurable function f : XxY [0, ∞0], √xy ƒ d(μ × v) = x ( [f(x,y) dv (y)) dµ(x). 2. Fubini's Theorem for Integrable Functions: Extend the result to integrable functions f€ L¹(XxY, xv), ensuring that the iterated integrals are equal and finite: fd(xv) = ( f(x, y) dv (y)) du (z dp(x) = √(√ f(x, y) dµ(z) dv (y). 3. Tonelli's Theorem: Prove that if f: X x YR is measurable and f(x, y) > 0 for all (z, y), then the iterated integrals are equal to the integral over the product space.

A' Reference book: KREYSZIG B[a,b] Introductory Functional Analysis with Applications Complement of a set A 18, 609 Transpose of a matrix A 113 Space of bounded functions 228 Space of bounded function 11 Space of functions of bounded variation 226 Space of bounded linear operators 118 B(A) BV4, b B(X, Y) B() Open ball 18 Closed bull 18 e A sequence space 34 C C[a, b] став CCX, Y) A sequence spuce 70 Complex plane of the field of complex numbers 6, 51 Unitary-space 6 Space of continuous functions 7 Space of continuously differentiuble functions 110 Space of compact linear operators 411 Domain of an operator 83 Dimension of a space X 54 Kroaecker delta 114 Spectral family 494 Norm of a bounded linear functional 104 Graph of an operator T 292 Identity operator 84 Infimum (greatest lower bound) 619 (T) 2(x,y) Distance from a toy 3 dim X b 2-(E₁) In (T) 1 inf Lab 1 r L(X, Y) M (T) " и A function space 62 A sequence space 11 A sequence space 6 A space of linear operators 118 Annihilator of a set M 148 Null space of an operator T 83 Zero operator 84 Empty set 609 1.3-1 Definition (Ball and sphere). Given a point xoeX and a real number r>0, we define three types of sets: (a) B(xo; ) {xEX | d(x, xo)0 there is a 8>0 such that (see Fig. 6) d(Tx, Txo)<ε for all x satisfying d(x, xo)<8. T is said to be continuous if it is continuous at every point of X. ■ 1.3-4 Theorem (Continuous mapping). A mapping T of a metric space X into a metric space Y is continuous if and only if the inverse image of any open subset of Y is an open subset of X. All the required definitions and theorems are attached, now need solution to given questions, stop copy pasting anything, I want fresh correct solutions, if you do not know the answers Problem 5: Measure-Theoretic Integration on Product Spaces Statement: Let (X, Ex.) and (Y, Ey,) be a-finite measure spaces. Consider the product a-algebra Σ. & Σ. on. Χ Χ Υ and the product measure μ. Χ. ν. Tasks: 1. Fubini's Theorem for Non-Negative Functions: Prove that for any non-negative measurable function f : XxY [0, ∞0], √xy ƒ d(μ × v) = x ( [f(x,y) dv (y)) dµ(x). 2. Fubini's Theorem for Integrable Functions: Extend the result to integrable functions f€ L¹(XxY, xv), ensuring that the iterated integrals are equal and finite: fd(xv) = ( f(x, y) dv (y)) du (z dp(x) = √(√ f(x, y) dµ(z) dv (y). 3. Tonelli's Theorem: Prove that if f: X x YR is measurable and f(x, y) > 0 for all (z, y), then the iterated integrals are equal to the integral over the product space.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.9: Properties Of Determinants

Problem 37E

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