Let X = Rand def ine ||.|| on X as ||x|| = 1 %3D then X is not normed linear space True O False O

Q: 4:: what the Null Space of is T;: R3 R³( T,x= xx a) R, R (T = X.Q

A: As per guidelines, we will solve first three sub-question: We have to find the null space of  1.…

Q: Let X be a normed space and aɛ X. Show that there exists an fe X' such that ||a|| = Sup { \f(a)| :…

A: Given that,                X is a normed space and a ∈ X . We have to show that there exists an f ∈…

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: (2.3.15. Let X be a metric space. (a) Prove that SCX is bounded if and only if diam(S) < 0o.

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

Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = Ā♂ (A) = A - int A (ii)…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: H is a Hilbert space, A: H → H is linear ar = for all r, y e H. Use th closed graph theorem to show…

A: For two normed spaces V and W, a linear operator T:V→W is closed if for each xn∈V, xn→x and Txn→y…

Q: _I. Let H and K be Hilbert spaces and A E B(H, K). Show that A is compact if = and only if A* A is…

A:

Q: Let X and U be Banach spaces and let C: X → U be a compact linear map. Assume that U is infinite…

A: Given that X and Y be Banach spaces and let C:X→U be a compact linear map. and let U is infinite…

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

A:

Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then bos nogo 2i (i) Fr(A) = A n (A)° A –…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Date. AC) Lat normed linear space X, Y= spomE and aex only if fla) = 0 wheneres f EX' omd f30…

A: To show the necessary part- To show that for each f∈X' whenever f vanishes on E, i.e.if f=0 then…

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: Note: Unless otherwise stated, E stands for a locally convex space (LCS). 2.4.6. Suppose p and q are…

A: Given that E is a locally  convex space also given that p and q are seminorms on a  Locally convex…

Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A n (A)° = A – int A…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Evaluat..f.x.dA. wha.is.the..sgiem.in.the.fiigt. R quandrant..bounded..xy=16. X..8....y.z.. y.e.x...

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Q: Let X = Rand def ine || || on X as |a|= min{|a,0) %3D then show that if X is a normed linear space…

A:

Q: Let D a ci+0 +1, t and f: D e fa) = + e 1 -2) Que fion 1 The function fe) is entiren True r False…

A: First find the pole. Then use the property of pole's.

Q: Suppose V is a vector space and S, T = L(V, V) are such that Range Scker T=null T. Prove that (ST)²…

A:

Q: Let X and Y be normed linear spaces, and T E B(X,Y). Prove that 38 > 0 such that 8||x|| < ||T(æ)||,…

A: Consider the given information.

Q: Let X = R and de f ine ||.|| on X as ||a|| = Va2 %3D then X is normed linear space O True O False

A: The given norm is Euclidean norm.

Q: Let A: C [0,1]→C [0,1]. Prove that operator continuous reversible, find A. 2.11. Ax(t)= x(1)- -…

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty. Prove that if s ϵ K,…

A: Given: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty The objective is…

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: Let X = Rand de f ine ||| on X as ||r|| = Va then X is normed linear space

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Q: Let X be a normed space and A € BL(X) be of finite rank. Then σe (A) = σ₁(A) = o(A).

A: 1

Q: 9. Let be a bounded open subset of C, and 4: Prove that if there exists a point zo E such that 2 a…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Let A = Z. Let R be defined on A x A where a Ry if 4 (x- y). Prove that R is transitive.

A:

Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A (A)° = A - int A %3D…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A O (A)° = A – int A %3D…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Q1) Prove or Disprove: Let L(F) be a linear space and A,B are two convex subsets of L. Then, AXB is…

A: The problem is related with convex set. Given that,                 LF  is a linear space  and A, B…

Q: E be a subset of a normed space X and Y = Span E and a € X. Show that Let a € Ỹ if and only if f(a)…

A: To find: a∈Y¯, and f(a) = 0 whenever f∈Xι and f is 0 everywhere on E.

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: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A. Ax() = x() + ja-m}xt…

A: Solution:- We have given operator A such that A: C[0,1]- C[0, 1] Also we must know that the inverse…

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…

A:

Q: Every T2 (Hausdorf f) space is T1 space. True O False O

A: Since T2 is a product preserving topological property. So T2 space is a T1 space.

Q: et H be a HIlbert space and S b of S is a closed linear space of

A:

Q: Theorem [4.1.11] Let A and B be subsets of topological spaces X and Y respectively. Then Int (A x B)…

A:

Q: 9) Suppose V is a vector space and S,T E L(V,V) are such that Range S c ker T =null T. Prove that…

A: See the detailed solution below.

Q: 2. Let f and g both be of bounded variation on [a, b]. Prove that this inequality holds. Vrag SV+V

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Q: 7. Let (S, p) be a metric space and T : S → S be a contraction mapping. Prove that T is uniformly…

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Q: Question 2: ( {(x,2x,3x):x E R}. Let E %3D (a) Show that E is not bounded. (b) Is E is compact? Why?

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Q: 4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = 0.

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

Q: Iw (X, D) discrete space A = {2,3,4s A A Bd(A), Euct CA)? EXercise %3D Fined

A: Given that A=2,3,4 The aim is to find A,A°, Bd(A), Eucd(A) A=2,3,4 And, A°=∅ Here A is closure of A…

Q: (b) space and X+{ 0 }. Show that B< (X, Y) is a Banach space. Let X be a normed space and Y be a…

A: X be a normed space and Y be a Banach space We have to show that B(X, Y) - set of all boundes linear…

Q: 3.7.5. Given a finite-dimensional normed space X, prove the following state- ments. (a) X is…

A: Multipart :    Let X be finite dimensional normed space. (a) We need to show that X is separable. As…

Q: A Consider E = \R as a vector space. Show that: if and only if fixed real winber A function f: E →…

A:

Q: Use th properties of linear Space A upon t to prove s a) 0.X - o for all X EX b) a.x=0 =>(a=07v…

A: