Instructions to follow: * Give original work Chatgpt means downvote, *Support your work with examples and graphs where required *Follow The references: Kreyszig, Rudin and Robert. G. Bartle. Reference Books: C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, (2000) J. Bak and D.J. Newman, Complex Analysis, 2nd Edition, Springer Indian Reprint, (2009) Bartle and Sherbert, Introductory Real Analysis, 3rd edition, Wiley International, (2001) E. Kreyssig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let KR or K C. We use the symbol, for example, f(x)= r² to say that the function f is defined by setting f(x) = for all in the domain. This is same as writing f(x) 2. Can you guess what the symbol 2 LIIS RIIS means that IIIS is defined by LIIS. f(x) means? I started with the principle that a first course in functional analysis is meant first as a part of the general culture and second as an important tool for any future analyst. llence the emphasis all through had been to look at concrete spaces of function and linear maps between them. This has two advantages: (1) the students get to see the typical applications of the results of functional analysis to other parts of analysis and (2) while dealing with such 16. Measure Theory - Fubini's Theorem and Product Spaces Let (X, A, μ) and (Y, B, v) be σ-finite measure spaces. Prove Fubini's Theorem, which states that iff: XxYR is a measurable function that is integrable on X x Y, then + Y) √xxy f(x,y) d(µ × 1)(x, y) = ſx (ſ, ƒ (x,y) dv(y) du(x) = f (ſx ƒ (x,y) du(x) dv(y). Provide a complete proof, covering all necessary steps, including the case for non-negative functions and extending to integrable functions. Hint: Structure the proof by considering simple functions and then applying the Monotone Convergence Theorem. 17. Operator Theory - The Spectral Radius Formula for Banach Algebras Let A be an element of a unital Banach algebra. Prove the spectral radius formula: (A) im ||A||³/ TX where r(A) is the spectral radius of A. Additionally, prove that the spectrum o(A) is a non-empty, compact subset of the complex plane.

A: Matrix of Linear Transformation: Suppose V,W are vector spaces with ordered bases B={v1​,...,vn​}…

A: Required codes:import numpy as np import matplotlib.pyplot as plt # Define the unit circle theta =…

A: We can solve the problem using the combination method in discrete mathematics.We want to choose 12…

A: The property that states v + w = w + v is known as the Commutative Property of Addition.

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A: 1.We are givenP=[−11​−12​]We use the following relationship for square matrix of order…

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A: Problem 1We are given two functions:f:Z→Z defined by f(x)=x2+x+1g:Z→Z defined by g(x)=2−5xWe need to…

A: Given:m = 8 lb 13 oz = 8.813 lb  (Mass of the shot put)R = 71 ft 8 in = 71.667 ft   (Horizontal…

A: To prove by induction that, for all n∈N and for all x∈R such that x>−1:(1+x)n≥nx Step 1: Base…

A: Definition: A graph is connected if there exists a path between every pair of nodes. In graph A,…

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A: Use this python code for graphs:import numpy as np import matplotlib.pyplot as plt # Create a grid…

A: (a) Average Rate of ChangeThe average rate of change of V(t) over a time interval [a, b] is given…

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A: Detailed Proof Using the Closed Graph TheoremLet H and K be Hilbert spaces, and let T: H → K be a…

A: 1.GivenT(x,y)=(x+y,3y)andB′={(−4,1),(1,−1)}The matrix A′ for R relative to the basis B′ is given by…

A: Required code:# Define the sequence x_n = (1/n, 0) in ℝ² n_values = np.arange(1, 21) # n from 1 to…

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A: Step 1: Eliminate the first element in the second row: R2 = R2 - 2*R1 1-j  1  -20  -j  1-j0  2+j…

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A: To show that H={1,−1} is a normal subgroup of the quaternion group Q, we need to verify that H is…

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A: For Question 1Step 1: Matrix AA :R3→R4 maps the input layer (3 nodes) to the hidden layer (4…

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A: Step 1: Identify what each symbol meansA ∩ B = A and B = intersection of A and BA ∩ B ∩ C = A and B…

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A: Exercise 2.3.4: Problem:Let (X, dX), (Y, dY), and (Z, dZ) be metric spaces. Let f: X -> Y and g:…

A: Part 1: The Mittag-Leffler function is a generalization of the exponential function and is defined…

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A: Required code:import numpy as np import matplotlib.pyplot as plt # Define the functions f_n(x) =…

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A: Use this code:# Visualization of functions in W^{1,2}(0,1) and their dual elements # Defining…

A: Step 1: Understand the properties of planar and non-planar graphsPlanar graphs are graphs that can…

A: RK4 Method ApplicationDifferential equationThe RK4 method approximates the solution at successive…

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A: Detailed Explanation: a. Submodules of M are Finitely Generate Because R is a commutative Noetherian…

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A: Required python code:# Define the example function u(x) = x(1 - x) and its derivative x_vals =…

A: Let's solve these contour integrals using well-known results from complex analysis, particularly…

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A: Approach to solving the question: In propositional logic, Well-Formed Formulae (WFF) are valid…

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A: To determine if the data provides convincing evidence that the population proportion of cats…

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