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

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: If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensional

A: This is the question of Functional Analysis.

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 V be a real inner product space, and let u, v, w E V. If (u, v) = 1/2 and (v, w) = -2, what is…

A:

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

A:

Q: * Every Pre-Hilbert space is I - Continues Normed space -orthogonal Metric space inner Produced

A: Every pre-Hilbert Space is a complex vector space X together with an inner product (A complex inner…

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: 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: 5. Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norm of function…

A:

Q: Find three different bases for the 3-dimensional real space (ususally denoted R3).

A: Basis: Basis of a vector space (V) over a field F (R or C) is linearly independent subset of vector…

Q: determine whether the set, together with the standard operations, is a vector space. If it is not,…

A: The given set is:0abc0def0

Q: Every inner product space is metric space and normed space O metric space but may be not normed O…

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

A:

Q: snip

A: Theorem:                  Suppose <v,w> is an inner product on a vector space V.  Then…

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: 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: 24 W is a vector space and chec ot a vector space because it do a vector space because it is r ector…

A:

Q: Determine whether the set, together with the standard operations, is a vector space. If it is not,…

A:

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

A:

Q: 2. Show that if the orthogonal dimension of a Hilbert space H is finite, it equals the dimension of…

A: Given: Finite dimension To find: Former finite

Q: Consider the spaceZ+with the finite complement topology. Is thisspace a Hausdorff space? Is this…

A:

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

A: This is a problem of Functional Analysis.

Q: ": Prove that ( R", ||||) is complete metric 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: ovide an illustration of the statement "Every normal space need not be a ular space."

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

A:

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

A:

Q: Let N and N' be normed linear spaces with the

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: Let X be a normed linear space such that its dual X' is separable, then show that X is separable ?…

A: This is a problem of Functional Analysis.

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: Determine whether the set, together with the indicated operations, is a vector space. If it is not,…

A: Consider the given set x,y:x≥2,y≥8

Q: Show that a space X is Hausdorff if and only if the diagonal {(x, x)|x € X} is closed in X2.

A: We use basic topological definition to prove this.

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

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…

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

A:

Q: Let H be a subspace of real space R. In terms of a geometrical definition of H, which of the…

A: Given:

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

A: