(b) Prove that the unit ball of a normed linear space is compact if and only if the normed linear space is finite dimensional. Use this to show that the identity map on an infinite dimensional normed space is not compact.

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 be a locally compact Hausdorff space which is not compact; Let X* be one-point compactificatin…

A:

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

A:

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

A:

Q: 6. Show that a compact metric space X is locally compact.

A: The given problem is related with metric space. We have to show that a compact metric space X is…

Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

Q: Prove that any Space is complete. finite dimensional normed

A: Introduction: Completeness is a property of a topological space. When in a topological space, every…

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

A:

Q: Let be a compact Hausdorff space, then find the necessary and sufficient condition for space to be…

A: Given that a compact Hausdorff space  To Find : Necessary and Sufficient condition for space to be…

Q: Prove that a unit ball Bx of a normed vector space X is compact if and only if X is finite…

A: Prove that a unit ball Bx of a normed vector space X is compact if and only if X is finite…

Q: let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B…

A: Let (X,t)  be  not  connected.We  have  to  show  that  X=A∪B ,where  A and B are  elements  in t…

Q: Which of the listed mappings is an isomorphism of vector spaces?

A: (a) A:R3→R3​[x],A:(a,b,c)↦ax+bx2+cx3 Linearity:…

Q: Let V be a C-vector space of dimension n, that is, dim(V) = n. Show that V is an R-vector space of…

A:

Q: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: Check whether the identity map on an infinite dimensional space is compact.

A: Identity map on a infinite dimensional space is not compact. We can show this with the help of a…

Q: Check whether a finite dimensional normed linear space is reflexive? Justify your answer.

A: A normed space X is called reflexive if it satisfies the following equivalent conditions: (1) the…

Q: Show that every normed liner space which is linearly isometric with a separable space is separable.

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

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: (a) Prove that the metric space (2,11) is complete. (b) Show that the metric space (Q.H) is not…

A:

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: Use F.Riesz Lemma to show that the closed unit ball in an infinite dimensional Banach space is not…

A: Let us prove above result

Q: a) Let V(R) be the inner product space and a, B E V Il a +B 12+| a - B 1?= 2(1| a |l² +I B I²) prove…

A:

Q: (e) Show that a compact metric space can be covered by finitely many balls of any given radius. (f)…

A: A compact space has the property that every open cover has a finite subcover. In the context of…

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: Show that a finite union of compact subspaces of a topological spaces X is compact-

A: We have to show that the finite union of two compact subspaces of a topological space X is compact.

Q: / Let (X₁T) and (Y₁J) be top. Spaces and fix-x be a surjective and continuous map, then y is Compact…

A:

Q: If A, B are subsets of the topological space X show that Ext(AUB) = Ext(A)n Ext(B).

A: Given: A and B are subsets of the topological space X. To Prove: ExtA∪B=ExtA∩ExtB

Q: (b) Let X be a normed space and Y be a Banach space. Define the operator norm on the linear space…

A: (b) It is given that X is a normed space and Y is a Banach space. Firstly, define the BX,Y  as,…

Q: Prove that any finite dimensional normed space space is complete-

A:

Q: prove that any finite dimensional normed space is reflexive.

A: Let us suppose that we have a finite-dimensional normed space, X For any natural number, n, let…

Q: 3: Prove that any reflexive normed space is complete.

A:

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

A: Solution

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: Show that a normed vector space is finite dimensional if and only if its closed unit ball is…

A: A normed vector space is a vector space over the real or complex numbers, together with a function…

Q: that Let f: X\rightarrow{}Y be a continues function from a compact topological space X into a…

A:

Q: f(z)=3iz

A:

Q: Once you get away from vector spaces of finite dimension, it is no longer true that every linear map…

A: As per bartleby guidelines for more than one questions asked only first should be answered . Please…

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…