A' Reference book: KREYSZIG B[a,b] B(A) BV1a,b BOX, Y) B(A) B(A) Ca C C" C[a, b] C[Q] C(X, Y) (T) Introductory Functional Analysis with Applications Complement of a set A 18, 609 Transpose of a matrix A 113 Space of bounded function 228 Space of bounded functions 111 Space of functions of bounded variation 226 Space of bounded lineur operators 118 Open ball 18 Closed ball 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 differentiuble functions 110 Space of compact linear operators 411 Domain of an operator 83 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 d(x, y) Distance from x toy 3 dim X бус 2-(E) М TH ' inf ["[a b] " A sequence space 11 F L(X, Y) M NIT) И Identity operator 84 Infimum (greatest lower bound) 619 A function space 62 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 xoЄ X and a real number r>0, we define three types of sets: (a) B(xo; r)=(xX | d(x, xo)0 there is a 8>0 such that (see Fig. 6) d(Tx, Txo)
A' Reference book: KREYSZIG B[a,b] B(A) BV1a,b BOX, Y) B(A) B(A) Ca C C" C[a, b] C[Q] C(X, Y) (T) Introductory Functional Analysis with Applications Complement of a set A 18, 609 Transpose of a matrix A 113 Space of bounded function 228 Space of bounded functions 111 Space of functions of bounded variation 226 Space of bounded lineur operators 118 Open ball 18 Closed ball 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 differentiuble functions 110 Space of compact linear operators 411 Domain of an operator 83 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 d(x, y) Distance from x toy 3 dim X бус 2-(E) М TH ' inf ["[a b] " A sequence space 11 F L(X, Y) M NIT) И Identity operator 84 Infimum (greatest lower bound) 619 A function space 62 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 xoЄ X and a real number r>0, we define three types of sets: (a) B(xo; r)=(xX | d(x, xo)0 there is a 8>0 such that (see Fig. 6) d(Tx, Txo)
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 44E
Related questions
Question
![A'
Reference book:
KREYSZIG
B[a,b]
B(A)
BV1a,b
BOX, Y)
B(A)
B(A)
Ca
C
C"
C[a, b]
C[Q]
C(X, Y)
(T)
Introductory
Functional
Analysis with
Applications
Complement of a set A 18, 609
Transpose of a matrix A 113
Space of bounded function 228
Space of bounded functions 111
Space of functions of bounded variation 226
Space of bounded lineur operators 118
Open ball 18
Closed ball 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 differentiuble functions 110
Space of compact linear operators 411
Domain of an operator 83
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
d(x,
y)
Distance from x toy 3
dim X
бус
2-(E)
М
TH
'
inf
["[a b]
"
A sequence space 11
F
L(X, Y)
M
NIT)
И
Identity operator 84
Infimum (greatest lower bound) 619
A function space 62
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 xoЄ X and a real
number r>0, we define three types of sets:
(a)
B(xo; r)=(xX | d(x, xo)<r}
(Open ball)
(1) (b)
B(xo; r) {xEX | d(x, xo)≤r}
(Closed ball)
(c)
S(xo; r) ={xx|d(x, xo)=r}
(Sphere)
In all three cases, xo is called the center and r the radius.
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 ɛ>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.
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 2: Radon-Nikodym Theorem in Abstract Measure Spaces
Statement:
Let (X,,) be a a-finite measure space, and let v be a a-finite measure on (X,Σ) absolutely
continuous with respect to μ(ie., v<<μ).
Tasks:
1. Construction of the Radon-Nikodym Derivative:
• Prove the existence of a measurable function f: X→ [0, ∞) such that for all A € Σ,
v(A)= ƒ dμ
f du.
2. Uniqueness Almost Everywhere:
Show that the function f is unique μ-almost everywhere.
3. Properties of the Derivative:
Prove that if and are probability measures, then f is in L¹(X,) and satisfies
Jx f du=1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2802fe82-f96c-4a87-93af-5ea2695b0fe4%2F85420feb-4b22-4ddf-8f36-59d75d470703%2Fbrvpl6f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A'
Reference book:
KREYSZIG
B[a,b]
B(A)
BV1a,b
BOX, Y)
B(A)
B(A)
Ca
C
C"
C[a, b]
C[Q]
C(X, Y)
(T)
Introductory
Functional
Analysis with
Applications
Complement of a set A 18, 609
Transpose of a matrix A 113
Space of bounded function 228
Space of bounded functions 111
Space of functions of bounded variation 226
Space of bounded lineur operators 118
Open ball 18
Closed ball 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 differentiuble functions 110
Space of compact linear operators 411
Domain of an operator 83
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
d(x,
y)
Distance from x toy 3
dim X
бус
2-(E)
М
TH
'
inf
["[a b]
"
A sequence space 11
F
L(X, Y)
M
NIT)
И
Identity operator 84
Infimum (greatest lower bound) 619
A function space 62
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 xoЄ X and a real
number r>0, we define three types of sets:
(a)
B(xo; r)=(xX | d(x, xo)<r}
(Open ball)
(1) (b)
B(xo; r) {xEX | d(x, xo)≤r}
(Closed ball)
(c)
S(xo; r) ={xx|d(x, xo)=r}
(Sphere)
In all three cases, xo is called the center and r the radius.
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 ɛ>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.
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 2: Radon-Nikodym Theorem in Abstract Measure Spaces
Statement:
Let (X,,) be a a-finite measure space, and let v be a a-finite measure on (X,Σ) absolutely
continuous with respect to μ(ie., v<<μ).
Tasks:
1. Construction of the Radon-Nikodym Derivative:
• Prove the existence of a measurable function f: X→ [0, ∞) such that for all A € Σ,
v(A)= ƒ dμ
f du.
2. Uniqueness Almost Everywhere:
Show that the function f is unique μ-almost everywhere.
3. Properties of the Derivative:
Prove that if and are probability measures, then f is in L¹(X,) and satisfies
Jx f du=1.
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