If X is finite dimensional hor med space o T: X-Dy is a Linear operator from X to normed space y. show that T is continue

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: 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: The domain D(T) is a vector space

A: Given, the domain D(T) is a vector space__________

Q: Is the set V = {(x, y) x >0} where (x1, y₁) + (x2, Y₂) = (x₁x2, y₁ + y₂) and c (x, y) = (x,cy) a…

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Q: H.W: Let V = R2, defined the addition by: (u1 +u2) + (V1 +v2)= ( U1 +V1 , U2 +V2 + 1) With standard…

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Q: Let S:x→Y be bounded lineay a operator, Then prove that ker S is closed.

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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: 10. Suppose that U and V are finite dimensional vector spaces and that F:VU is a linear map. prove…

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Q: THEOREM 4.4 (LINEAR TRANSLATION) Let Y= aX + b; then: My(0) = eb" My(a0).

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Q: Consider the vector space R³ and d₂(x, y) = (-1lx - y²). Show that d₂ satisfy M3 (The tringle…

A: Consider the vector space We  have  to prove triangle  inequality   that  is  we  have to how that…

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 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: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: Show that the spaces (R,U) and (R,S) are Hausdorff.

A: Let a, b be any two distinct points of R so that a ≠ b

Q: defined as Let V = {(x,y) | x, y = R}, with vector addition and scalar multiplication (x₁, y₁) (x2,…

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Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V × V → V…

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Q: Ex3 i) ii) Let T: C ([0,1]) → R f → f(0) If C ([0,1]) is equipped with the sup-norm, show that T is…

A: As per the policy, we are solving first question. Please repost it and specify which question is to…

Q: Let V = F" where FE {C, R} show that V is a Banach space with the norm ||x| = maxisisn{xil VxEF".

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Q: 1. Let V = C[-2, 2], the vector space of continuous functions on the closed interval [-2, 2). Let f,…

A: We can just multiply h and k and integrate  Then after simplifying we are getting answer

Q: Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive…

A: Below given clear explanation

Q: Say the linear operator given T:R³ → R³, defined by T(x, y, z) = (x + y, x – y + 2z, 2x + y – 2). I…

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

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Q: Let V = {(x,y) where y= 3x+1 x ER} with scalar multiplication defined by k(x,y)=(kx,y) is a vector…

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Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V x V → V…

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Q: 16) Let V be the vector space of functionsf:R R.Let S be the subspace generated by fe,e, 2x -2x X…

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Q: 3. Show that the vector space Cla, b] of all continuous complex valued functions defined on [a, b],…

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Q: If W is a subspace of R" and ax is in W, where a is a nonzero scalar and x is in R". Then x is in W/…

A: We have to check

Q: a: Let T be a linear operator on a finite dimensional vector space V. Suppose T is diagonalizable.…

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Q: Is every liner operator T : IR2 → IR2 diagonalizable? Justify!

A: Linear transformation and diagonalization

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

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Q: 6. Let T is a linear operator on a finite-dimensional vector space V. Prove that V R(T) + N(T)…

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