(a) Define a reflexive space. Prove that a Banach space is reflexive if and only if its dual space is reflexive. (b) State closed graph theorem. Deduce open mapping theorem from the closed graph theorem.

Q: A topological space is Hausdorff iff limits of all nets in it are unique

A: This is a problem of Topology.

Q: 0:1 What are the basic properties of finite dimensional space, explain that briefly.

A: We are supposed to answer only one question.

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Q: what is reflexive Banach space ?

A: Banach space: A normed linear space N,· is called banach space if it is complete metric space. A…

Q: 3. is Show R with explamtion. Banach space, a

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A: This is a problem of topology.

Q: Prove that coo cannot be a Banach space under any norm.

A: To prove: c00 cannot be a Banach space under any norm.

Q: Theorem 2.26. Let A be a subset of a topological space X. Then p is an interior point of A if and…

A: First well show that, if there existsan open set U with P∈UCA, thenP will be an interior point of…

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Q: Define a compact space.

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

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A: We have to give an example for an open set A ⊂ R with the following property: "The boundary of the…

Q: Define a bijective map g : x2/3--->R.

A: Sol

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Q: Prove that linear maps are bounded.

A: To prove that linear maps are bounded.

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

Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

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Q: At least five example of affine mapping?

A: At least five example of affine mapping?

Q: topological space a

A: Presumably, the closure of Q in R under the usual topology. It suffices to show that for every real…

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Q: how to prove a map is unique?

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Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the…

A: Given : X be a set with atleast two points. To prove : Discrete space and indiscrete space ie. (X,τ)…

Q: The homeomorphic relation between two. spaces is not reflexive O an equivalence relation O not…

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Q: Prove that continuous image of pathwise connected space is pathwise connected

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Show that if a Banach space X is non-trivial, X + {0}, 3). then X' # {0}.

A: According to the given information, it is required to show that:

Q: Wueslien no 0BN Banach space 3. Show that R is a

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

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Q: Prove: If F is a closed subset of a compact space X, then F is also compact.

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Q: Prove or give counter example an infinite product of discrete spaces may not be discrete

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Q: Prove that the image of a continuous mapping of a compact set is compact.

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Q: Can you show an example of Null space?

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Q: 3. а) When is a normed linear space called separable? Show that a normed linear space is separable…

A: a. Let X be a normed linear space. Then, X is called separable if there is a countable dense subset…

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Q: The intersection of any countable family of open dense-sets in a locally compact space X is dense.

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Q: Quertion:- Show that R is a Banach Space.

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Q: what ordered pair is an example of a topological space that is connected but not path-connected,why?

A: We use contradiction method.

Q: The closure of a compact subject of a regular space is compact.

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Q: Suppose X is a topological space, Y is a Hausdorff space, and f:X→Y is continuous. Show that the…

A: Suppose X is a topological space, Y is a Hausdorff space, and f:X→Y is continuous. Show that the…

Q: A discrete metric space X is separable if and only if X is countable

A: Given: To prove: A discrete metric space X is separable if and only if X is countable

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Q: 2.2 Is any discrete space complete? Give an example to substantiate your answer.

A: Yes, any discrete space is complete.

Q: (a) Give a bounded linear non zero map that is not an open map. Does this contradict the open…

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Q: Prove that every normed space is continuous and moreover uniformly continuous function. Give an…

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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: Show that the closure of a connected space is connected.

A:

Q: ive example to show that a quotient space of a second countable space need not be second countabl

A: This is a problem of topology.

Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.