(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: The set of vectors in R? with y = -3x is a real vector space

A:

Q: Use Hahn-Banach Theorem to show that a normed linear space X is finite dimensional if its dual X' is…

A:

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: Prove directly that the dual space of a normed space is complete.

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: every metric space is normed space? give an example to show tha

A: We have to show that every metric  space  may not be a  normed  space.

Q: Define a compact space.

A: The objective is to give the definition of a compact space.

Q: Show that W is a vector space.

A:

Q: The shape's vector set is a subspace of R3

A: solution :-

Q: Prove that a vector space is infinite-dimensional if and only if it contains an infinite linearly…

A: Given the statement that a vector space is infinite-dimensional if and only if it contains an…

Q: - theorm; – Every inner product space is a norm linear space

A: We have to prove that every inner product space is a norm linear space.

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: d) The set of real numbers, with addition defined by x + y %3D х - у

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: ive an example of a space is not vector space

A:

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: Define a reflexive normed linear space. Prove that if a normed space X is reflexive then its dual…

A: A Banach space X that coincides under the canonical imbedding with its second dual X∗∗ (cf. Adjoint…

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

A:

Q: (b) Suppose -|, and ||-||2 are two equivalen norms on a vector space X. If (X, ||.||,) is complete,…

A:

Q: Does the empty set form a vector space?

A:

Q: Prove the locally compact normal linear space is finite dimensions.

A: We have to prove the  locally compact normal linear space is finite dimensions First we define the…

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 or disprove every metric space is normed space.

A: Statement :-  Prove or disprove every metric space is normed space.

Q: Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1…

A:

Q: Prove that a closed linear subspace of a reflexive space is reflexive.

A: This is a problem of functional analysis.

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

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: (a) Show that every metric space is T2-space.

A:

Q: one example of a 4-dimensional vector space that is not R 4

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: 1 x dy -) (3y+--- x dx 2 y 1 y x +(52+-- y = 0

A:

Q: A subspace of a completely regular space is completely regular.

A: