Let f: X → Y be a continuous map from a compact space X to a Hausdorff space Y. Let C be a closed subspace of Y, and let U be an open neighborhood of f-¹(C) in X. Show that there is an open neighborhood V of C in Y such that f-¹(V) is contained in U.

Q: Show that V is not a subspace of R°: V is the set of all (x, y, z) such that x + y +z = 3

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Q: The set of all continuous real-valued functions on an interval (a,b) is denoted by C(a,b) and is a…

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Q: Let X = {x, y, z} and B = {{x, z}, {x}, {y}}. Let T be the topology generated by B and consider B =…

A: Given: T is topology generated by B=x, z, x, y and B=x, z is a subspace of a topological space (X,…

Q: Does the function d: Y x Y have the same three properties? Explain why or why not.

A: We have to find the function d: Y x Y have the same three properties? Explain why or why not.

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: Show that [0,∞) and (0,∞) as subspaces of R with the usual topology are not homeomorphic.

A: Given: [0, ∞) and (0, ∞) are subspaces of ℝ. To show: [0, ∞) and (0, ∞)are not homeomorphic.

Q: A function f: R → R is called even if f(-x) = f(x) for all x ER, and a function f: R → R is called…

A: Given that Ue denotes the set of all real valued even functions and Uo denotes the set of all real…

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: M2x2 (R) is a vector space over R. Make use of this result to show that the subset S defined by a S…

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Q: (b) Let f be a bounded linear functional on a subspace M of a normed space X and let X, E X, x, 4 M.…

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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: Let V be the vector space of real-valued functions on R. Let H be the set of all linear equations;…

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

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Q: 2. A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is…

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Q: If V is a vector space over F of dimension 5 and U and W are subspacesof V of dimension 3, prove…

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Q: Consider the subspaces U = {(x,x) | x ≤ R} and W = {(0, y) | y ≤ R} of R². Show that U + W is a…

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Q: Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V such…

A: Let V be an finite dimensional inner product space and U and W be subspace of V and U∩W=0¯ We have…

Q: Let V be an inner product space with inner product (, ) and let W be a subset of V. Show that the…

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Q: Let M be a closed linear subspace of a Banach space X. Prove that the quotient space X/M, with the…

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Q: b) Check whether the sequence (5,5,4,4,3,3,3, 1,1, 1) is graphic or not. If it is graphic, draw a…

A: Given the sequence (5,5,4,4,3,3,3,1,1,1) We need to check whether it is graphical or not. If its…

Q: a) ir(A - 1)1- ( -2) %3D Find the matrix A

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Q: Let V, W be vector spaces over F and let T:V → W be a linear map. Prove that if T is injective, then…

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Q: Consider the set W={(x, y, x+y) | x, y are real}. Is Wa subspace of R³? If so, prove it. Otherwise…

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Q: Let V be finite dimensional vector space and let W c V be subspace with complimentary subspace W'.…

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Q: Let E, be a closed subspace and E, be a finite dimensional subspace of a normed subspace X. Prove…

A: This is a question of Functional Analysis.

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: be

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Q: (c) Let X be a Banach space and let Y be a closed subspace of X. Let n : X → X/Y be. the Canonical…

A: Given: X is Banach space and Y is closed subspace of X. Let π:X→X/Y be the canonical quotient map.…

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: Let V be an n-dimensional vector space. We call a subspace of dimension n-1 a hyperplane. a) If : VF…

A: Given that V is a n-dimensional vector space. We know that a subspace of dimension n-1 is called ah…

Q: Let (R, Tus) be the real line.Let r ∈ R such that 0 ≤ r < 1 and consider I = [0, 1] as a subspace of…

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Q: Let V be a vector space of finite dimension n over a field F, and let U be a subspace of V of…

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Q: Let V = {(x,y) | x,y ϵ R} – the set of ordered pairs in R² . Let W = {(x,0) | x ϵ R}  Prove that W…

A: The objective is to prove that W is a subspace of V.

Q: 6. Let V be a vector space with subspaces U and W. Show that the set UnW = {x|x EU and xEW} is a…

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Q: Consider the subspace W {(x, y, z) : x = 2y and x, y, z E F} of F³. Are F² and W isomorphic? Either…

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