the euclidean space R³ is a separable metric space. true or false?

Q: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

A:

Q: Let (R, d) be usual metric space, then Ris ot separable True False

A:

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

A:

Q: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

A:

Q: Let (R. d) be discrete metric space then R is not complete O True O Faise

A:

Q: Let (X, d) be a metric space and let A ⊆ X be complete. Show that A is closed.

A: Given that X,d be a metric space and A⊆X be complete. The objective is to show that A is closed.

Q: メ=D¢ c omplex number %3D Doyou CGid) is metric space of not ??

A:

Q: Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded

A:

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

A:

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

A:

Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

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

Q: Define metric space and define its four properties

A:

Q: Every metric space is a hausdorff space .

A:

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

A: Answer is mentioned below

Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

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

A:

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

A:

Q: Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M);…

A: Given as X,T be a topological space,M,d be a complete metric space and BCX,M=f∈CX,M;fX is bounded…

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: Show that R is a bianch space.

A:

Q: Prove that any reflexive normed space is complete

A: A normed space X will be said to be complete if each Cauchy sequence in X is convergent in X.

Q: If S be a n-dimen Sional normed Space, then prove that its dual space is al so n- dimen Sional .

A: A normed space E is called a Banach space if it is complete, i.e., if every Cauchy sequence is…

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: Let X and Y be compact spaces. Then X × Y is compact.

A:

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

A:

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

A:

Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

A:

Q: What is Euclidean Space??

A:

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

A:

Q: A of a metric space (x, d.)

A:

Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

A:

Q: Find all limit points of a set A = {(2+,2 – 1)|n e N} on the Euclidean plane with the usual metric.…

A:

Q: ovide an illustration of the statement "Every normal space need not be a ular space."

A:

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

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: Prove that Let X be a metric space such that X E1 U E2. If E1, E2 are complete, then X is complete.

A:

Q: Task 13 Let (M, d) be a compact metric space. Show that closed subsets of M are compact.

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

A:

Q: Every inner product space is a metric space O True O False

A: TRUE

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

A:

Q: Show that (C(X),|| · ||∞,x) is a metric space.

A:

Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: Every T2 (Hausdor ff) space is T1 space. True O False O

A:

Q: Give the definition of a separable metric space (S, d). Show that the metric space (IR, d), where…

A:

Q: For a normed space X, prove that the dual of X is separable implies that X is separable. Is the…

A: NOTE: According to bartleby we have to answer only first question please upload the question…

Q: Let E and F be compact sets in a metric space (X,d). Show that E U F is compact using the definition…

A:

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

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…

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

A:

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.