let X be a firite dimensional notm Space Then Prove that is compat.

Q: ) Consider the sequen ces equi' pped space f absolutely ce uith He Ofalloing' onvergent norm (0.1)…

A: We will use the basic knowledge of norms and functional analysis to answer all three parts of this…

Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norn of function f(x)…

A: This is a problem of functional analysis.

Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

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: Let (R. d) be discrete metric space, then Ris separ able O True O False

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Q: A ° = A – JA

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Q: (a) Suppose that (M, d) is a metric space. Consider the mapping f: M [0, 1] be given and let di(x,…

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Q: Let X = Rand def ine |-|| on X as ||r||= V then X is not normed linear space O True False

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Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: A completely regular space is also T1-space 1 True 2 False

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Q: 3. Let Ac R' where A = {x: x 1}. Describe the quotient space topology on R'/A obtained by…

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Q: be nonempty Subsets of Space X. Then Prove Let A and BJA inner product an TT 10) A CAL TT %3D (C) A…

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Q: Let X = R and def ine ||.|| on X as ||x|| = x² then X is normed linear space True False

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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: If M is a closed linear subs r space X, and x, be a vect the distance from x, to M, ional f, in X'…

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Q: Let (X, d) be a metric space, and p: X→ IR doxy) I+desy) de fined by puM pusy) = %3D show that p is…

A: Consider a set X and a function defined from it to the real numbers as d : X×X→ℝ. Then, X,d is a…

Q: rove that spaces Ip for os p<l If for pe(0, 0) {0} and complet. are

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Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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Q: Letxandy and z be normed space s and s: xy andTIYz bounded oper atorsprove

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Q: Let (x, g) be a metric space. Is the function d (x, y) = k.g (x, y) (k € R) defined from XxX to R a…

A: Let X,g be a metric space . Given that function X×X to R defined as  dx,y=k·gx,y where k∈R A metric…

Q: Assume that (M, d) is a compact metric space. Show that if f: (M,d) → (Y, d) is continuous and…

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Q: If S be a n-dimen Sional normed Space, then prove that its dual space is al so n- dimen Sional .

A: A normed space E is called a Banach space if it is complete, i.e., if every Cauchy sequence is…

Q: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

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Q: ) Let X be a finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

A: Sufficient to prove that M is closed and bounded

Q: Let V be a real inner product space, and let u, v, w E V. If (u, v) = 1 and (v, w) = 3, what is (3u…

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Q: Let (X, E, µ) be a measure space. Show that f is integrable if and only if fx ]f]dµ < o∞. Moreover…

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Q: Consider the inner product space R" with the usual dot product. Which of the following sets is NOT…

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Q: Let X be the subset (R x {0}) U (R x {0}) CR2 Show that the quotient space X/~ is locally Eue line…

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Q: Let X be a finite dimensional normed space (a) and A :X→X is a linear map. Show that A is compact.…

A: Given: X is a finite dimensional normed space and A : X → X is a linear map. We have to show that A…

Q: / Let (X₁T) and (Y₁J) be top. Spaces and fix-x be a surjective and continuous map, then y is Compact…

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Q: * Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((r, y)) = x…

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Q: let a-Rbe ametric sPacesConsir he following subsed ofP @ Show that Dra De are comPact Setsornot of D…

A: Question from real analysis.

Q: Use the Completion Theorem below to show that the Completion of the discrete metric space X is…

A: By completion theorem, there exists a complete metric space which has a subspace that is isometric…

Q: metric

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Q: Show that it A is subset of X any with the discre te metric then %3D

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Q: Let (X, T) and (Y, T1) be topological spaces and f a mapping of X into Y. If (X, T) is a discrete…

A: For proving f is continuous enough to prove that inverse image of every open set is open.

Q: Let A be a sSubset of a metric space A. Given r€A, define the distar from x to A to be dist(z, A) =…

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Q: If the unit sphere {xe X : x = 1} in a normed space X is complete, prove that x is complete.

A: Given that X is a normed space. Let S1x=x∈X| x=1 be unit sphere of the normed space X. Given that…

Q: Let X,d) be a metrie space. Define d': Xx X >R by: d'Cxoy) Show that (X.d') is min (dl8), 1). a…

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Q: Q7: Prove that ( IR", II-||) is complete metric space.

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

A: If X and Y both are Banach spaces then we need to prove that X⊕Y is Banach space with respect with…

Q: Every inner product space is a metric space O True O False

A: TRUE

Q: tet is cosinte toploy on set R show that (R, t) is O NOVmal space- Regualra space

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Q: Show that (C(X),|| · ||∞,x) is a metric space.

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Q: Let X be a compact subset of R and let C(X) denote the set of continuous real-valued functions on X…

A: let X be a compact subset of ℝ and let CX denote the set of continuous real valued functions on X.…

Q: Prove the parallelogram law on an inner product space V; that is, showthatllx + Yll2 + llx - Yll2 =…

A: Prove the parallelogram law on an inner product space V; that is,  Show that…

Q: 3 \et F be anormid space and- |2+j|l <llxll+yllprove that F has a stridtl (3) y convex- orm

A: We know that a normed space is strictly convex if and only if x≠y, ∥x∥=∥y∥=1, we have ∥x+y∥&lt;2 Let…

Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…

Q: Let (X, || · ||) be a normed linear space. Prove that (a) B1(0) = B1(0).

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