Prove the following: The countable product of metrizable spaces is a metrizable space.

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

A:

Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: a) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

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

A:

Q: Every metric space is a hausdorff space .

A:

Q: A metric space is sequentially compact if and only if it is totally bounded and complete. prove it

A: We need to prove that any given metric space is sequentially compact if and only if it is totally…

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: 36) Prove that every product Space is inner metric Space- Discuss its Converse

A: Suppose that V is an inner product space over a field F. To prove that the V  is a metric space. To…

Q: (a) Prove that every finite subset of a metric space is closed and has umulation points.

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

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: Every T3 space is regular space. True False

A: Since a regular T1 space is called a T3 space.

Q: Any cover of a metric space is open.

A: Any cover of a metric space is open_____?_____

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

A:

Q: Prove that: 1. (R, |. |) is a complete metric space. 2. (Q.I-1) is not complete metric 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: Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

A:

Q: Every finite space is: (a) Compact space (b) Not compact space (c) Hausdorff space. (d) Non of above

A:

Q: Is the following statement True or False? Justify each answer. If S is a closed bounded subset of a…

A:

Q: every finite subset of metric space is compact. ture or false?

A:

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

A:

Q: Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

A: Given:  (Y,d) is a metric space, and Y is totally bounded To Prove Y is bounded Definition: Let…

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

A:

Q: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

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: 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: 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: If X is an inner product space and A = {0}, then A = %3D %3D O X O A

A: A⊥ = X

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

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

A: TRUE

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: Every closed subset of a connected metric space is connected. True False

A:

Q: Theorem 9.8. A metric space is Hausdorff, regular, and normal.

A: The objective is to show that every metric is hausdorff, regular and normal. Let x≠y be points of a…

Q: Remark: If (V, = « +p vi " = a + p

A:

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: 2) Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

A:

Q: (a) Prove that the metric space (Z,) is complete.

A: Complete Metric Spaces: A metric space (X,d) is said to be complete iff every Cauchy sequence in X…