Reference book: 4° A' Bab B(A) BV, KREYSZIG B(X, Y) B(r) B(A) Co C C" C[a, b] Club) C(X, Y) 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 functions 11 Space of functions of bounded variation 226 Space of bounded linear operators 118 Open ball 18 Closed bull 18 A sequence space 34 A sequence space 70 Complex plane or the field of complex numbers 6, 51 Unitary-space 6 Space of continuous functions 7 Space of continuously differentiable functions 110 Space of compact linear operators 411 Domain of an operator T83 d(x, y) Distance from a toy 3 dim X 6x 2-(E) 6(T) Dimension of a space X 54 Kronecker 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 A function space 62 [[a b] A sequence space 11 r L(X, Y) M (T) " и 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 xo EX and a real number r>0, we define three types of sets: (a) B(xo; r) {xe X d(x, xo)0 there is a 8>0 such that (see Fig. 6) d(Tx, Txo)< 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 just skip it otherwise I will downvote you. Problem 1: Spectral Theorem for Compact Self-Adjoint Operators on Hilbert Spaces Statement: Let H be an infinite-dimensional separable Hilbert space over C, and let T: H→ H tea compact, self-adjoint operator. Tasks: 1. Eigenvalue Sequence Construction •Prove that the non-zero spectrum of T consists of a sequence {A} C IR converging to zero. Show that each A, has finite multiplicity. 2. Orthonormal Basis of Eigenvectors: •Demonstrate that there exists an orthonormal basis {e} for H consisting of eigenvectors of 1 3. Spectral Representation. Establish that I can be represented as 00 T-E) -1 where the series converges in the operator norm topology.

Reference book: 4° A' Bab B(A) BV, KREYSZIG B(X, Y) B(r) B(A) Co C C" C[a, b] Club) C(X, Y) 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 functions 11 Space of functions of bounded variation 226 Space of bounded linear operators 118 Open ball 18 Closed bull 18 A sequence space 34 A sequence space 70 Complex plane or the field of complex numbers 6, 51 Unitary-space 6 Space of continuous functions 7 Space of continuously differentiable functions 110 Space of compact linear operators 411 Domain of an operator T83 d(x, y) Distance from a toy 3 dim X 6x 2-(E) 6(T) Dimension of a space X 54 Kronecker 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 A function space 62 [[a b] A sequence space 11 r L(X, Y) M (T) " и 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 xo EX and a real number r>0, we define three types of sets: (a) B(xo; r) {xe X d(x, xo)0 there is a 8>0 such that (see Fig. 6) d(Tx, Txo)< 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 just skip it otherwise I will downvote you. Problem 1: Spectral Theorem for Compact Self-Adjoint Operators on Hilbert Spaces Statement: Let H be an infinite-dimensional separable Hilbert space over C, and let T: H→ H tea compact, self-adjoint operator. Tasks: 1. Eigenvalue Sequence Construction •Prove that the non-zero spectrum of T consists of a sequence {A} C IR converging to zero. Show that each A, has finite multiplicity. 2. Orthonormal Basis of Eigenvectors: •Demonstrate that there exists an orthonormal basis {e} for H consisting of eigenvectors of 1 3. Spectral Representation. Establish that I can be represented as 00 T-E) -1 where the series converges in the operator norm topology.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.5: Iterative Methods For Computing Eigenvalues

Problem 51EQ

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