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

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Q: where a, b e X, is also a metric on X. 2- Let (X, d) be a metric space and T the corresponding…

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Q: Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric…

A: A matric space X is separable if it has a countable dense subset.

Q: Thm. Let v be an inner product space. and жуєх V then ||_ [121]. [171] further equility is hold iff…

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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: 2. Are the following sets real vector spaces? Justify your answers.

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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: Do the following task. a)  State formally what do we mean by being a metric space of X with metric…

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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: (b) Let (M, d) be a metric space. Define a function e: MXM-R by e(x, y) = min {1,d(x, y)}; vx,y E M.…

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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: Assume that (M, d) is a compact metric space. Show that if f: (M,d) → (Y, d) is continuous and…

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Q: Let (X, T1) and (Y, T2) be topological spaces. Show that the set {U × Y ∣ U ∈ T1} ∪ {X × V ∣ V ∈…

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Q: (a) Show that d(m, n) = |m²-n²| 1+ m²-n²| defines a metric on the set of natural numbers N. Hint:…

A: Given That: d(m,n)=|m2-n2|1+|m2-n2|   To Show : a) d  defines a metric on the set of natural number…

Q: LA3-2. Suppose f is a nonzero linear function on some vector space V and U = ker f. Prove that V = U…

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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 (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If…

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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: H.W: Let V = {(x, y): x 20} with standard operation on R2. Is Va vector space?

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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: Is the following a vector space? Let 'V' = 'R'. Defined for 'j', 'k' residing in 'V' and 'm'…

A: Given that V=R, defined for j,k residing in V and m residing in R for the following operations:…

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

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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: This picture shows a set of points in the plane R. Which of the following axioms of a vector space…

A: As negative values are not there in the graph . So, additive inverse does not  exist.

Q: Exc.: Let c[a, b] = (f:f: [a, b] → R be a cont. function), define d: c[a, b] * c[a, b] → R as d(f,g)…

A: Let Ca,b=f:a,b→ℝ be continious function  define d: Ca,b∗Ca,b→ℝ as df,g=∫abfx-gxdx

Q: Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i.…

A: To verify that given distance form d is a metric , we have to check for  Identity Symmetry…

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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Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given Data : Suppose H is the set of all points x , y in R2 such that x2+xy +3y2=3. To Show : H is a…