8- Suppose that X,Y be a normed spaces then the function f: X → Y is continuous at a point xo E X if and only if f is a sequentially continuous at the same point. True O False

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A: Ck=set of all functions that are continuous and have first k derivatives that are continuous To show…

Q: 4. Suppose that (X, dx) and (Y, dy) are metric spaces, and define distance on X × Y by Prove that…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: (2.9.9 Let X and Y be metric spaces. Show that a function f: X Y is continuous if and only if f-'(E)…

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Q: Let f be a continuous real-valued function defined on a metric space (X, d), and let D be a compact…

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Q: If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensional

A: This is the question of Functional Analysis.

Q: Let A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

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Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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Q: 4. Use the definition of uniformly continuity to prove that f (x) 1- x is uniformly continuous on…

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Q: This is a real analysis question. Suppose that (X,d) and (Y,ρ) are metric spaces. (i)The function…

A: We will solve both these parts (a) and (b) as requested by you, using just the given definitions of…

Q: If a subadditive functional p on a normed space X is continuous at 0 and p(0) = 0, show that p is…

A: Given: A sub additive functional p on a normed space X is continuous at 0 and p0=0. To prove: p is…

Q: Prove the Theorem 5.5.7 - Let (X1, d1) and (X2, d2) be metric spaces and let f : X1 → X2. Then f is…

A: Continuity on a metric space is very significant. Continuity on a metric space is related to open…

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: 2) let X and y be metric spaces and T. XY Y a ContinUOus mapping Then the image of a Compact subset…

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Q: 3.3.9. Given a normed space X, prove that the function p: X R defined by p(x) = ||x|| is continuous…

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Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded. %3D

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Q: (b) If (X ,7) is a topological space, then the inclusion function i :X →X where i (x ) =x for each x…

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Q: 2. If X,Y are Banach spaces on XOY define |(x, y) = |, +|yly . Show that this gives a norm making X…

A: Given: Dual of c0  is isometric to l1 State the principle of uniform boundedness.

Q: Prove that coo cannot be a Banach space under any norm.

A: To prove: c00 cannot be a Banach space under any norm.

Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: [0,1] 6. If f is continuous on • prove that

A: We can solve an integral of some function f(x) w.r.t. x by using substituting, say, t=f(x) which…

Q: Let (R. d) be discrete metric space, thend can not induces a norm on X O True O Fahe

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

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Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…

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Q: 3. Let X= {a,b,c} with and r, = {ø, X,{a,b},{b},{b,c}} and Y ={1,2,3} with topology T, =…

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Q: Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c)…

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Q: 4. Prove that a function which is harmonic everywhere on a plane and is bounded either above or…

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Q: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: 2. (a) Let f be defined for all x, y by f(x, y) x-y - x. Show that f is concave (i) by using Theorem…

A: Given: a) f(x, y) =x - y - x2 Theorem 2.3.1: This theorem says that if f is twice continuously…

Q: [0, 4). 4. Prove that f(x) = x² is both uniformly continuous and continuous on

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Q: If f:X - Y is a continuous function then f- (V) is open in X for every open set V in Y. True False

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Q: 4. Prove fx=x2+4 is not uniformly continuous on R

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Q: let (2, m) be a meesurable space. then 1f f9: x LJare measural ble function, then -> ) f43 is mea…

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Q: 8- Let X, Y are metric spaces and f: X → Y .If for any A C X, f(A) C f(A) then f is continuous.

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Q: Theorem 8.9. Let f : X Y be a continuous, surjective function. If X is connected, then Y is…

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Q: Let X be a Banach space and let (T,} be a sequence of bounded linear operators on X such that Tx=…

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Q: 2. (a) Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let X be a topological space with the property that, for every topological space Y, every function f…

A: The space X with the discrete topology means every singleton of space is open.

Q: 3.27.2. Let X be a metric space with metric d; let ACX be nonempty. (a) Show that d(x, A) = 0 iff x…

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Q: a) Show that, any function f: X → Y is continuous if X have discrete topology.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 13- Suppose that W is a subspace of normed space V then it is not necessary that W is also normed…

A: Subspace of a normed space

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Q: 5) Let X and Y be normed spaces. Suppose T:X → Y be a closed linear operator. Let (xn) be a sequence…

A: We will prove this by using definition

Q: (a) Let X and Y be normed spaces and F :X→Y be a continuous linear map. Show that F is closed . Is…

A: Given that,                   X and Y are normed spaces and F : X → Y is a continuous linear map. We…

Q: 2.9.11 Prove that every function f: Z→ F is continuous, where the metric on Z is the restriction of…

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Q: Q3: A) If f is continuous function on a compact metric space X Show that £ieunifone

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Q: 2. Define f : (2,7) → R by f (x) = x³ – x + 1. Use the definition of uniformly continuity to prove…

A: Consider the given information. fx=x3-x+1 As the function is in the form of a polynomial.  Now, find…

Q: 2.2 Let s = {(x,y) ER² :x,y2 1} be a subset of R2. Define f:S→R by f(x,y)= ÷- The determine if f is…

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Q: Theorem 4.12: If f: X → Y and g: Y → Z are continuous functions on the indicated spaces, then the…

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