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: Let V be a finite-dimensional inner product space over C, and let S,T be linear maps V + V. (i)…

A: 1) Adjoint Map: Let V be an inner product space and T:V→V be a linear map. A map T*:V→V is called an…

Q: Let fg,:I→R2 begivenby f(x) = (0, x), g(x) = (1, x). a. Specify a homotopy h:I × I→R2 from f to g.…

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Q: How do you know that for all a, b, c, d: ax + bx will be orthogonal to cx? + d ?

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Q: Show that the operator U is a partial isometry U: L² ([0,1]) → L² ([0,1]), Uƒ(x) = X[0,1] ƒ(x)

A: To show that the operator U:L2([0,1])×[0,1]→L2([0,1]) defined by Uf(x)=χ[0,1](x)⋅f(x) is a partial…

Q: (a) Prove the set (A(*)./a(*),fa(a)) is linearly independent. (b) Use Gram-Schmit process to find an…

A: Given,     f1(x)=3,  f2(x)=1-x, and f3(x)=x2+2x-1

Q: Let C = = {0, 1,2,3} and f: C→C, g: C→C with f = {(0, 2), (1, 2), (2, 3), (3, 0)}, = {(0, 3), (1,…

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Q: 4. Suppose that T : R* → R² is a linear map defined by T (x1, x2, x3, x4) = (2x1 x2 + x3, -2x1 – x2…

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Q: Let L: R→ R be a linear operator, if L(X)= L(Y), then X= Y. Select one: O True O False

A: Given Linear operator is  L:R3→R3 we have to check if L(X)=L(Y) the X=Y

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) =…

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Q: Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive…

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Q: Prove that if X is a Banach space, X ∕= {0}, then there is a non-zero biounded linear functional on…

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Q: Let X and Y be topological spaces and endow X×Y and Y×X with the respective product topologies. Show…

A: To Show : Let X and Y be topological spaces and endow X×Y and Y×X with the respective product…

Q: For all v € V, let : V' → F be defined by (p) = y(v) for y € V'. (a) Show that is linear, i.e. € V",…

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

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Q: 2. - For each map below, either prove that it is linear or demonstrate that it is not linear. (a) T…

A: Consider a map T:U→V is linear if    (i) T(u+v) =T(u)+T(v)   (ii) T(αu )   = αT(u)   , for all u,v…

Q: Given points p = and p3 = in R?, 2 P2 = let S = conv {p],P2» P3}. For each linear functional f, find…

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Q: 11) Let f: IR4 → IR³ be defined by : IV f(x₁y₁z₁t) = (2x+y-t, x+8y-92- 2t, x-2y + 3z) (a) show that…

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Q: Let p(x) = 2-x and q(x) = x. Using the inner product space = integral 1 to -1 pqdx find all…

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Q: Let L: R2 R³ be a linear mapping such that Find L -19 ([¹2]) -13 absin (a) 8 dr f 8 L D-||| (D-|| =…

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Q: . Let B(X, Y) denote the space of all bounded linear operators between the normed spaces X and Y.…

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

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Q: Let L: C[0, 1] → C[0, 1] be a linear map. If L(1) =x, L(x) =x^2, then what is L(3x + 2)?

A: Given that  be a linear map and .To find the value of .

Q: Consider X = [1,2]? with the relation (x1, y1)~(x2, Y2) if x1Y2 = x2y1. Also define 3x g: X → [1,2]…

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