In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and r> 0.

Q: If τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normal space…

A: τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normalspace or…

Q: The set ((x,y) : × 20, y 20) with the standard operations in R in not a vector space.

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Q: Prove in full detail that the set {(x, 2x): x is a real number}, with the standard operations in R2,…

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Q: Prove that the open ball B(a,r) is a subset of the interior of the closed ball C(a,r)

A: Given that Ba,r is an open ball of radius r and centered at a. Then a∈Ba,r                ...... (1)…

Q: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

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Q: A point y in space either belongs to a given closed c ough y so that whole of the x lies in one open…

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Q: Let V be an inner product space, and suppose that T: V→V is linear and that ||T(x)|| = ||x|| for all…

A: Given:

Q: Q3 Prove : If V is a normed linear space, and WcV, and U a closed subset of W Then given ve V,3u"…

A: given if V is a normed linear space, W⊂V and U is a closed subset of W then for v∈V there exist u*∈V…

Q: Let (X, d) be a metric space and let A ⊆ X be complete. Show that A is closed.

A: Given that X,d be a metric space and A⊆X be complete. The objective is to show that A is closed.

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

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Q: For a normed space X and a closed convex subset A of X. Prove that (º )nÎ= Â. A

A: Let, be a normed space and be the closed convex subset of As, we know that norm on is a function…

Q: The space X = {0,1} with topology T = {0, {1}, {0, 1}}) is To but not T Select one: O True O False

A: Discrete spaces are always T0 , So X is T0 .  A space X is a T1 space if and only if  for each x∈X,…

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

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Q: Suppose V is an inner-product space and u, v e V satisfy ||u|| = ||v|| = 1 and (u, v) = 1. Prove…

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Q: Exercise 8.2. Let X, Y, Z be normed spaces and suppose that T E B(X, Y) and S e B(Y, Z). Show that…

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Q: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

A: Given, X and Y are normed spaces and X ≠ 0. We have to prove that BLX , Y is a Banach space with the…

Q: Let ( Xr d be ame tric space,show that exist atopology on X induce b

A: Let (X, d) be a Metric Space We know that in a Metric Space  (1) arbitrary union of open sets is…

Q: Show that ℓ^1 is a normed linear space.

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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: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

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Q: Extend the linearly independent Set S={x+1, 2X²} to the space base P₂ (R)

A: We are given a linearly independent set S= {x+1, 2x2} We have to extend S to basis of P2(R).    Let…

Q: A. Let H be the set of all points (x, y) in R2 such that x2 + 3y2 = 12. Show that H is a closed…

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

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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: If A is 3 × 3 with rank A = 2, show that the dimension of the null space of A is 1.

A: Given: A is 3*3 matrix with rank A = 2

Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

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

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Q: A of a metric space (x, d.)

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Q: Let X be a topological space. Construct a topological space Px and a continuous surjection 0: X→ Px…

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

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Q: (b) Prove that the unit ball of a normed linear space is compact if and only if the normed linear…

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Q: Prove that a normed linear space is complete if its unit sphere is complete.

A: We prove that a normed linear space is complete if its unit sphere is complete. Its straightforward…

Q: Let E be a Euclidean space with inner ratic form. For all u, v E E, we have the C y(u, v)² ≤…

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Q: H.W.2 Exercise Prove that: 1. (IR, J- |) is a complete metric space. 2. (Q. |-1) is not complete…

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

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Q: Let (X, || · ||) be a Banach space, J CR be bounded and F C C1 (J, X ). If sup ||f(x)|| < , f€F, rɛJ…

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Q: Let X, Y be normed linear spaces. On X x Y define || (x,y) |l, = || × |lx + || y lly and + || y ily…

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Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

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Q: In each part, explain why the set S is not a spanning set for the vector space V. (a) S = {t³, t²,…

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Q: Let (X, d) be a compact metric space and T: XX be a mapping that satisfies d(Tr,Ty)<d(r,y) for all…

A: What is Metric Space: Metric space is an abstract set in mathematics, particularly topology, with a…

Q: 14. If d is a metric on a vector space X‡ {0} which is obtained from a norm, and å is defined by…

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Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…

Q: Let E and F be compact sets in a metric space (X,d). Show that E U F is compact using the definition…

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Q: If x is a normed linear space and the closed Unit ball M is compact, then X is finite dimensional.

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Q: Let X is a normed linear space and the ball M= {xe X:|| x ||≤1} is compact, then X is finite…

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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…