Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN: 9781133382119
Author: Swokowski
Publisher: Cengage
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[画像:A' Reference book: KREYSZIG Introductory Functional Analysis with Applications Complement of a set A 18, 609 Transpose of a matrix A 113 Space of bounded functions 225 Space of bounded functions 11 Space of functions of bounded variation 226 Space of bounded linear operators 118 B[a,b] B(A) BV4, B(X, Y) B(x) Open ball 18 B(x) Closed ball LR A sequence space 34 C C[a, b] Clab] C(X, Y) A sequence apuce 70 Complex plane or the field of complex numbers 6, 51 Unitary u-space 6 Space of continuous functions 7 Space of continuously differentiable functions 110 Space of compact linear operator 411 Domain of an operato T 83 Dimension of a space X 54 Spectral family 494 Norm of a bounded linear functional 104 Graph of an operator T 292 (T) d(x, y) Distance from toy 3 dim X Krouecker delta 114 2-(E) TH 1 inf ["[a b] " r L(X, Y) M N(T) И Identity operator 84 Infimum (greatest lower bound) 619 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 x, EX and a real number r>0, we define three types of sets: (a) B(xo; r) {xeX|d(x, xo)<r} (Open ball) (1) (b) B(xo; r)= {xX|d(x, xo)≤r} (Closed ball) (c) S(xo; r) ={xex | d(x, xo)=r} (Sphere) In all three cases, xo is called the center and r the radius. I 1.3-2 Definition (Open set, closed set). A subset M of a metric space X is said to be open if it contains a ball about each of its points. A subset K of X is said to be closed if its complement (in X) is open, that is, K-X-K is open. 1.3-3 Definition (Continuous mapping). Let X = (X, d) and Y = (Y, d) be metric spaces. A mapping T: XY is said to be continuous at a point x0 = X if for every >0 there is a 8>0 such that (see Fig. 6) d(Tx, Txo)<E for all x satisfying d(x, xo)<8. T is said to be continuous if it is continuous at every point of X. ■しかく 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 4: Unbounded Operators and Self-Adjoint Extensions Statement: Let H be a Hilbert space, and let T: D(T) CH→H be a densely defined, symmetric, unbounded linear operator. Tasks: 1. Adjoint Operator Existence: • Prove that the adjoint operator T" exists and that TCT". 2. Self-Adjoint Extensions via von Neumann's Theory: Using von Neumann's theory of deficiency indices, characterize when I admits self-adjoint extensions. Specifically, show that I has self-adjoint extensions if and only if the deficiency indices n+(T) = n(T). 3. Essential Self-Adjointness: •Provide conditions under which I' is essentially self-adjoint, i.e., T=T".]
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Transcribed Image Text:A' Reference book: KREYSZIG Introductory Functional Analysis with Applications Complement of a set A 18, 609 Transpose of a matrix A 113 Space of bounded functions 225 Space of bounded functions 11 Space of functions of bounded variation 226 Space of bounded linear operators 118 B[a,b] B(A) BV4, B(X, Y) B(x) Open ball 18 B(x) Closed ball LR A sequence space 34 C C[a, b] Clab] C(X, Y) A sequence apuce 70 Complex plane or the field of complex numbers 6, 51 Unitary u-space 6 Space of continuous functions 7 Space of continuously differentiable functions 110 Space of compact linear operator 411 Domain of an operato T 83 Dimension of a space X 54 Spectral family 494 Norm of a bounded linear functional 104 Graph of an operator T 292 (T) d(x, y) Distance from toy 3 dim X Krouecker delta 114 2-(E) TH 1 inf ["[a b] " r L(X, Y) M N(T) И Identity operator 84 Infimum (greatest lower bound) 619 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 x, EX and a real number r>0, we define three types of sets: (a) B(xo; r) {xeX|d(x, xo)<r} (Open ball) (1) (b) B(xo; r)= {xX|d(x, xo)≤r} (Closed ball) (c) S(xo; r) ={xex | d(x, xo)=r} (Sphere) In all three cases, xo is called the center and r the radius. I 1.3-2 Definition (Open set, closed set). A subset M of a metric space X is said to be open if it contains a ball about each of its points. A subset K of X is said to be closed if its complement (in X) is open, that is, K-X-K is open. 1.3-3 Definition (Continuous mapping). Let X = (X, d) and Y = (Y, d) be metric spaces. A mapping T: XY is said to be continuous at a point x0 = X if for every >0 there is a 8>0 such that (see Fig. 6) d(Tx, Txo)<E for all x satisfying d(x, xo)<8. T is said to be continuous if it is continuous at every point of X. ■しかく 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 4: Unbounded Operators and Self-Adjoint Extensions Statement: Let H be a Hilbert space, and let T: D(T) CH→H be a densely defined, symmetric, unbounded linear operator. Tasks: 1. Adjoint Operator Existence: • Prove that the adjoint operator T" exists and that TCT". 2. Self-Adjoint Extensions via von Neumann's Theory: Using von Neumann's theory of deficiency indices, characterize when I admits self-adjoint extensions. Specifically, show that I has self-adjoint extensions if and only if the deficiency indices n+(T) = n(T). 3. Essential Self-Adjointness: •Provide conditions under which I' is essentially self-adjoint, i.e., T=T".
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