For which of the following space the unit ball is weakly compact a) ℓ2; b) ℓ1; c) C[0, 1]; d) ℓ∞; e) L3[0, 1].

Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: Q3 Assume that (M,d) is a Compact metrie Space. Show that : (M,d) → (ya) is Continous bjective then…

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Q: 1. Let d, be a metries on Xo and let the function a: X x XR defined be dr. y) dis. y)= prove that…

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Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

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Q: . u. ): Is the set S = {r +x-1,22 - r+3,-a+3z-2}) a base for the Pa(r) vector space which is a…

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Q: The space X = {0,1} with topology T = {0, {1}, {0, 1}}) is To but not T Select one: O True O False

A: Discrete spaces are always T0 , So X is T0 .  A space X is a T1 space if and only if  for each x∈X,…

Q: Let (x,d) be metric space and A, BC X. 2. Show that (AnB)= A'n B and AUB = AUB %3D

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Q: Let (X, T) be a topological space and A,B CX- ConstYuct the general Yelation between Fr (IntlA) and…

A: For any subset B of a topological space X , Frontier or boundary of B is defined as  Fr(B) = CL(B) -…

Q: Show the 3-dimensional ball B₁: ((..) ER³ | x² + y² +2² 0.

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Q: Pyove that. the triangle axiom from the defination for a eqquivalent withe Convex:ty of the Unit…

A: Convex set: If for  then for all  we must have  then we can say that the set S is the convex set.…

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: (7) Which of the following is disconnected space: (a) (R\ {0}, Tusuat) (b) (R, Tusuat) (c) (X, t).…

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Q: Let V = R³ and W = {(a, b, c) E V | a² + b² = c?}. Is W a sub- space of V? If so, what is its…

A: As per the question we are given the three dimensional vector space V = ℝ3 and a subset W is defined…

Q: 8. Suppose I is a triangle formed by placing three points lircle, two of which lie on the circle's…

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Q: Q3 Assume that (M,d) is a Compact metric Space. Show that, f: (M,d) → (Y, 2) is Continous bjective…

A: f:(X, d) to (Y, d) is said to be homeomorphism if  1) f is continuous.  2) f is bijective.  3) f…

Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: 3. For which of the following space the unit ball a) (2; b) e1; c) C[0, 1); d) l; e) L³[0, 1]. In…

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Q: (a) Prove that the metric space (2,11) is complete,

A: Complete Metric Spaces:- A Metric Space (X,d) is said to be complete iff every Cauchy sequence in X…

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Let V is an inner product space. Prove, Va, y EV ||x+y||² + ||x − y||² = 2||x||² + 2||y||* . How…

A: To prove a result of the inner product space.

Q: Q7: Prove that ( IR", ||·|| ) is complete metric space.

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Q: Show that both ℝ and ℝ2 may be covered by countably many open spheres.

A: We have to show that  and  covered by countably many open balls.

Q: ) Give definition to the convex set of the space Rn. Also give a definition for when the function f…

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Q: Let (X, d) be a metric space. The diameter of a non-empty subset M of X is defined as 8(M)= sup d(x,…

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Q: Show that (Cod) is metric space

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Q: Let v= C[-1,1] What is the norm of the functionf(x) = - x in this space? O A) 2V5 O B) 2V10/5 OC)…

A: An inner product in the vector space of continuous functions in [-1, 1], denoted asV=C-1, 1, is…

Q: Which is true of the choices given for the R² space? A) R² = R B) R²=R×R ( x : cartesian product…

A: We have to check

Q: a) Find a space that is a regular but not Hausdorff. Justify your claim.

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