(b) Show that the metric space (Q,1) is not complete. (c) Prove that the discrete metric space (M,d) is complete.

Q: 1- a = (1, .,0) in the usual metric space R. 2- (0,1) in the usual metric space R. 3- {(x, y): xy =…

A: As per our company guidelines we are supposed to answer ?️only first 3 sub-parts. Kindly repost…

Q: Prove that any contraction between metric spaces has at most one fixed point

A: According to the given information, it is required to prove that any contraction between metric…

Q: Is the euclidean space R³ is a complete metric space

A: We have to check whether euclidean space R³ is a complete metric space or not .

Q: Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norn of function f(x)…

A: This is a problem of functional analysis.

Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

A:

Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: Exercise Prove that: 1. (R, J- |) is a complete metric space. 2. (Q. I-1) is not complete metric…

A:

Q: If X =R and d(x.y)30, then X metric space not metric semi metric

A:

Q: Exercise 2. Let X = C[0, 1], where C[0, 1] is the set of continuous functions on [0, 1], and the…

A: A set (X,d) is called a metric space if these four properties are satisfied for all x, y, z ∈ X: d…

Q: Let (X,d) be metric space and X is unbounded, then d (z. y) . y EX, is 1+d(r,y) di de fined as dy =…

A: Metric space

Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded. %3D

A:

Q: A completely regular space is also T1-space 1 True 2 False

A:

Q: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: . Prove or disprove every metric space is normed space,

A: A normed vector space or normed space is a vector space over the real or complex numbers, on which a…

Q: Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

A:

Q: If X is vector space and (X,d) is discrete metric space, then X can induce an inner product space…

A:

Q: Consider IR as a metric space.with metric dux, y) = 1x-y1. Show that the open interval C0, 1) is an…

A:

Q: Let (X, d) be a metric space, and p: X→ IR doxy) I+desy) de fined by puM pusy) = %3D show that p is…

A: Consider a set X and a function defined from it to the real numbers as d : X×X→ℝ. Then, X,d is a…

Q: Show that: All completions of a metric space are isometric.

A: According to the given information, it is required to show that all the completions of a metric…

Q: Show that ℓ^1 is a normed linear space.

A:

Q: Show that R is a bianch space.

A:

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: Prove that any linear map between finite dimensional normed linear spaces is continuous.

A:

Q: Theorem: Every normed vector space is a metric space but the converse is not true in general Proof

A:

Q: let X be a firite dimensional notm Space Then Prove that is compat.

A: As X is a finite dimensional norm space, this means that, dimX is finite.

Q: 3.4 Give the definition of a separable metric space (S, d). 3.5 Show that the metric space (R, d),…

A:

Q: Exercise. Prove that in any metric space (X, p), a closed ball {r E X : p(a, x) <r} is closed.

A:

Q: 1. Show that the real line is a metric space.

A:

Q: Let X be a finite dimensional normed space (a) and A :X→X is a linear map. Show that A is compact.…

A: Given: X is a finite dimensional normed space and A : X → X is a linear map. We have to show that A…

Q: Prove that a linear map on a normed vector space is bounded if and only if it is continuous.

A:

Q: 0:1 Show that norm ||x|| = (Elx¡l?)ž satisfies the conditions of normed space. %3D

A:

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

A:

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

A:

Q: The metric subspace ]1,2] of the Euclidean metric space R is a complete metric space. False True

A: Solution

Q: Prove that a finite dimensional normed linear space is always reflexive.

A:

Q: Define what it means for a topological space to be completely regular,

A:

Q: be a n-dimensional normed space, then prove that its dual space is also n-dimensional.

A: Let X be a normed linear space. Let X* be its dual space with the usual dual norm ||T|| = sup{…

Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

A:

Q: (b) Prove that the unit ball of a normed linear space is compact if and only if the normed linear…

A:

Q: 3 Let (Y, p) be a metric space. Give an example metric d on Y such that every finction f:(Y, d)→(Y,…

A:

Q: Prove that a normed linear space is complete if its unit sphere is complete.

A: We prove that a normed linear space is complete if its unit sphere is complete. Its straightforward…

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

A:

Q: If X is a metric space and Y C X, then Y is a metric space.

A:

Q: Exercise. Prove that in any metric space (X, p), a closed ball {x E X : p(a, x) <r} is closed.

A:

Q: Prove the parallelogram law on an inner product space V; that is, showthatllx + Yll2 + llx - Yll2 =…

A: Prove the parallelogram law on an inner product space V; that is,  Show that…

Q: Show that if X and Y Hausdorff spaces, then so is the product space X × Y. are

A:

Q: If x is a normed linear space and the closed Unit ball M is compact, then X is finite dimensional.

A:

Q: Show that the real number R with d(x,y)=|x-2y| are not a metric space

A:

Q: Let (X, || · ||) be a normed linear space. Prove that (a) B1(0) = B1(0).

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