6-4 - Let X and Y be normed spaces and Z be a closed subspace of X. (a) Let F E BL(X,Y) and Z C Z(F). The map F: X/Z → Y given by F(r + Z) = F(r), z € X, is well-defined, F E BL(X/Z,Y) and |F|=|FI. (b) If F E BL(X/Z,Y) and we let F(x) = F(x + Z) for x € X, then FE BL(X,Y) and ||F|| = ||F||. 6-5 Let X be a normed space and f be a nonzero linear functional on X. Then f is discontinuous if and only if Z(f) is dense in X. 6-6 Let Y

Q: (b) If V = U OW is a direct sum of subspaces U, W C V then any element v E V can be written uniquely…

A: Given V=U⊕W is a direct sum of subspaces U,W⊂V.  Then, it is known that any element v∈V can be…

Q: Consider L = I – A with A : X → X is a compact linear operator in a normed space X. Show that the…

A: The given problem is related with functional analysis. Given that L = I - A , where A : X → X is a…

Q: Let V be a vector space, and let W₁,. Let W = .... W₁ + + Wk = {w₁ + + wkw₁ € W₁, ..., wk € Wk}. (a)…

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Q: The set S of all functions f such that f'(1) = 0 is a subset of the vector space F of all functions…

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Q: Let U be a subspace of the vector space P100 (R), with dim(U) = 37. Suppose W is another subspace of…

A: P100(R) is a vector space  of all polynomial of degree 100 and smaller. The dimension is 101.For…

Q: Suppose V is a vector space with a non-trivial subspace W, i.e., W is not {0} and W is not V. If p :…

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

Q: • Let V = F(R, R) be the space of real-valued functions, and let C = {ƒ(x)|ƒ(0) = [ƒ(1)]²} Is C a…

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Q: Suppose Y is a closed subspace of a normed linear space X and xo ¤ Y, x, € X . Prove that there is a…

A: Y is a closed subspace of a normed linear space X and x0∉Y, x0∈X.. We have to prove that there is a…

Q: 27. Let W = sp(ex, ex, esx) be the subspace of the vector space of all real-valued functions with…

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Q: 8. Let V be a vector space over F and v₁, ₂, € V. Then (₁₁) = (a₁v₁ + + a₂v, la,,a,F) is a subspace…

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Q: If U is a subspace of V, show that T(U)= {T(u) u in U} is a subspace of W (called the image of U…

A: Let U be a subspace of V. We have to show that TU=Tu| u∈U is subspace of W.

Q: (3) The subset of pairs {(r, y) E R²|ry = 0} in the vector space R².

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Q: (a) F = Q, V = R², and W = Z². (b) F = R, V = R², and W = {(x, y) E R? : ry > 0}. (c) F = R, V = R³,…

A: Let W≠0 be a subset of a vector space V over F with the same addition and scalar multiplication as…

Q: (a) Let X be a normed linear space and M be a closed linear subspace of X. Let x, e X be such that…

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Q: Let P denote the vector space of polynomials (of any degree) and E P → R lenote the evaluation map…

A: Please find the solution in next step

Q: 5. Let V be the vector space of all continuous functions on [1,5]; that is, V = C[1,5]. -5 Let W be…

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Q: 7. Let S = {(3x² + 1, x − 1) | x € Z}. (a) Show that S is a subset of R. (b) Are (13,-3) and (7,1)…

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Q: 22. Let f be a linear functional on a subspace S of a finite-dimensional inner product space V. Let…

A: Please find the answer in next step

Q: i) Determine whether or not the set w - ((x.y.2) ej+y +-1) is a subspace of w i) Write (1,7,17) as a…

A: W is not subspace because (0,0,0) does not belongs to this set  ( 0^2 + 0^2 + 0^2 is not equal to 1)…

Q: ) Let VCR be a subspace. Let F: V→ V and G: V → V be invertible linear transformations. Denote by…

A: Kindly refer to Step 2 for the solution.

Q: (b) Let V be an n-dimensional F-vector space, and let 0 < k < n. If K is a k- dimensional subspace…

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Q: Prove or disprove: U = {ƒ €R[x] | f"(x) = 0} is a subspace of R[x].

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Q: be

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Q: 5. Let U = {(u₁, U2, U3) E R³|u₁ + u2 - u3 = 0} and V = {(V1, V2, V3) E R³v₁ - 3v2 +5v3 = 0}. (a)…

A: (a)

Q: The necessary and sufficient condition for a non to be a subspace of V is a - empty subset W of a…

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Q: W be a linear map. Let Vo 3 Let V andW be finite-dimensional vector spaces and T: V be a subspace of…

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Q: V 3 (R) is a vector space and W₁ = {(a, 0, c) : a, c = R} and W 2 = {(0, b, c) : b, c € are two…

A: Provided that V3 is three dimensional vector space over field Rand V has two W1 and W2 subspace…

Q: find the derivative of (2³ + j)² O 23 +j O 223 + j O 625 + j6z² O 325 + j3z?

A: Derivative of (z3+j)2 is 2(z3+j)2-1(3z2+0) = 2(z3+j)×3z2 = 6z2(z3+j) = 6z5+j6z2

Q: Let U be the set of all vectors (x1, x2, x3) satisfying 3x1 (x1, x2, x3) satisfying 2x₁ − x2 + x3 =…

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Q: (a.2) Let V be the vector space of polynomials over K of degree n-1 andf: V -> V the map defined by…

A: Given that V be the vector space of polynomials over K of degree less than equal to (n-1) and

Q: 5. Let P2(C) be the vector space of polynomials of degree less than or equal to 2 over C. Let C3 be…

A: 5) Let P2ℂ be the vector space of all polynomials of degree less than or equal to 2 over ℂ. and ℂ3…

Q: 5. Let V be a vector space, and let x1,X2, ... , Xk E V. 5(b) Prove that if x1, X2, ..., Xk is…

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Q: Let W = {(x, y, 1) | x, y, E R}. Is W a subspace of R3?

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Q: ts) Let P2 denote the vector space of polynomials of degree up to 2. Which of the following subsets…

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