5). Let 21, 22 be fixed elements of a Banach space X, and l₁, l2 € X'. Define A : X → X by Ax = l₁(x)z₁ +l₂(x)2₂. Show that A is compact.

Q: if and only if F-1: R(F) →X is bounded.

A:

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

A:

Q: Let X be a normed space and let 0x0 EX. Show that there is f = X* such that f(x) = 1 and ||f|| =…

A: Given, a normed space  and .It is required to show that, there exists  (the dual space of ) such…

Q: Let X and Y be topological spaces and fix some yo E Y. Show that the map f: X→XXY defined by f(x) =…

A:

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

A:

Q: Let (X,7) be a topological space and ACX. Show that (n) A is open if and only if Int(A) = A. (b) A…

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

Q: Let X be a topological space

A:

Q: Let X = C[0, 1] be the Banach space endowed with the sup-norm. Define the operator T: X→R by Tf := f…

A:

Q: let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B…

A: Let (X,t)  be  not  connected.We  have  to  show  that  X=A∪B ,where  A and B are  elements  in t…

Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

A:

Q: Suppose A is a function that assigns to each subset of the plane a number in such a way that the…

A: Given that AX≥0 and if X and Y are disjoints then AX∪Y=AX+AY To prove if X⊂Y then AX≤AY It is known…

Q: Show that the set A = {fm,b : R → R | m # 0 and fm,b(æ) group under composition of functions. (The…

A: To Show- Show that the set A = fm,b : R → R | m ≠ 0 and fm,bx = mx + b, m, b ∈ R forms a group under…

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 X, Y be the non-zero normed spaces. Then for any non-zero element x = X, there is a bounded…

A: Let's construct the linear map T:X→Y described as follows.Since x is a non-zero element of X, let's…

Q: Let T X Y and G: Y→ Z be the bounded linear operators between normed spaces. Show that the…

A:

Q: if (X,d) is discrete metric space, ACX and A+D (x,d) Find the sets of A° and A (functional analysis…

A: Consider a discrete metric space X,d where d is defined as: dp,q=0,p=q1,p≠q

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: 1. Let V be an inner product space. Suppose f,g E V such that f -gg (a) Prove that 1I2 = |lf- gll2 +…

A: To prove the required norm property in an inner product space, under the given conditions

Q: Q2:- Let (X, d) be a metric space. Xo, Yo EX and r, s > 0. Show that B, (yo) Br(xo) if s+d(xo, yo)…

A:

Q: Show that for any space A there does not exist an injection P(A) → A.

A: To prove that for any space A, there doesn't exist any injection PA→A Let us consider the contrary.…

Q: Let fbe a continuous function on an open subset U of a Banach spe into a Banach space Y. Let a and b…

A:

Q: 2.1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B R by…

A:

Q: 2. Let (,): P3 × P3 → R defined by (p, q) = p(0)q(0) + 2p'(0)q'(0) + 3p"(0)q"(0) + p''(0)q"'(0) for…

A: Given, ,:ℙ3×ℙ3→ℝ defined by, p,q=p(0)q(0)+2p'(0)q'(0)+3p''(0)q''(0)+p'''(0)q'''(0)  ∀p,q∈ℙ3 a) To…

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 Y be an infinite subset of a compact set XC R. Prove Y' 0.

A:

Q: Let f be a continuous functio to R. Let a and b be two distinct ntained in U and fbe differentiable…

A:

Q: Find the maximum of 6x? + 17y* on the subset of R? consisting of those points (x, y) such that (x –…

A:

Q: Prove that if A−1exists, then there can be no nonzero y for which Ay = 0.

A: Prove that if A−1exists, then there can be no nonzero y for which Ay = 0.

Q: 1. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is…

A:

Q: 12. Suppose that a, b E R and a < b. Show that the vector space C[a, b] of all continuous complex…

A:

Q: 2. Let [0, 1] = {ƒ : [0, 1] → R : f is a bounded function on [0, 1]}. Let ||f||∞ := := sup f(x)|…

A:

Q: 1. Let A, B be any sets with power sets P(A) and P(B), let AC P(A) and BC P(B), and let f : A →→ B.…

A:

Q: 4. Let C₁ be the set of all continuous real-valued functions on I = [0, 1]. (a) Show that d : C₂ ×…

A: According to our guidelines I can answer only Question no. 4(a).

Q: Exercise 3. 1) Let (X, d) be a metric space. Show that f X → R is Lipschitz with Lipsch constant L…

A:

Q: - Let A = {1, 2, 3, .., 103. Consider f:P(A) → N given by f(B) = |B|. That is, f takes B subset of A…

A:

Q: We note that ℤ2 which means ℤ*ℤ. Let d: ℤ2 *ℤ2→ℝ≥0 be the function that is given by…

A: The given in this question is a function d that takes two points (x0,y0) and (x1,y1) in the set of…

Q: Let X be a normed linear space and let P, Q E B(X). Show that the transformation T: B(X) B(X)…

A:

Q: Let A C Rª be a convex set. f : A → R is a strictly quasi-concave function if and only if for any a…

A:

Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: 10. Let X be a compact space and let Y be a Hausdorff space. Prove that X the projection Ty : X × Y…

A: We are given that X is a compact space and Y is a Hausdorff space. We need to prove that the…

Q: R and C with ||x|| = |x| are Banach spaces.

A: Given: ℝ and ℂ with x=x.  We have to prove that they are Banach spaces.

Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

A:

Q: Suppose that f1 : [0, 1] → R and f2 : [0, 1] → R are continuous everywhere and that f1(0) f2(1).…

A: Intermediate value theorem : Suppose be a continuous function then it takes all values between .