Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an L-Lipschitz function. Then there is an L-Lipschitz extension. 3.

Q: Let Φ be a set of isometries of a space A. Three distinct points a, b, c ∈ A we call “Φ-collinear”…

A: Sol:- (a) For any three distinct points a, b, c in E2, we can always find an isometry T that fixes a…

Q: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

A: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

Q: Consider the function T:R² → C defined by T(x,y)=x+iy. Show that this is an isomorphism of real…

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Q: Let V be a finite dimensional vector space and suppose T, SE L(V) satisfy TS = ST. %3D Prove that,…

A: Let V be a finite dimensiona; vectorspace and suppose T,S ∈ L(V) satisfyTS=STprove thatdim…

Q: Consider the set U = {fe Co (R)| limo f(x) = 0}. That is, U is the set of all continuous functions…

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Q: Let D be a Euclidean domain with measure d. Prove that u is a unitin D if and only if d(u) = d(1).

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Q: Let (X, d) be a metric space. Define f: Xx X → R by f(x, y) = = that f is a metric on X. d(x, y) 1+…

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Q: 1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5.…

A: Let, f and g be two functions in V.  Then,        f"(x) = f(x) - 5       g"(x) = g(x) - 5.

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 vector space and suppose there exist maps S, T = L(V, V) such that range(S) ℃ null(T).…

A: Let V be a vector space and suppose there exist maps S,T:V→V such that .We need to prove that the…

Q: D: Remind that the set F of all functions from R to R is a vector space. 2. Prove that for any…

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Q: [1030] Let A = 0 120 %3D V be the null space of A, and W= V.What is the dimension of W? 0 0 00 1 3.…

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Q: Let V be an inner product space over F. (a) Let u, w e V. Prove that ||2+ (u, w)w|> |u+||(u, w)w|2,…

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Q: Let f function from (R,d) to (R,d1) defined as d(x) = x, where d is the usual metric and .di is the…

A: Metric space function

Q: Suppose W1, ..., Wm are vector spaces. Prove that L(V, W1 × • × Wm) and L(V, W1) × ·…× L(V, Wm) are…

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