Define what it means for a topological space to be completely regular,

Q: Date Find two examples for equivalont norms on three dimensional Yeal Space R3.

A: Question: Find two examples for equivalent norms on three dimensional real space ℝ3

Q: Q2) Let (A,T)be topological spaces; i=1,2,3. Then A, x A2 x Az is homomorphic to (A, x A2) x A3 I

A: Definition : Let X and Y be spaces, and let B be the collection of sets U × V , where U ⊆ X and V ⊆…

Q: Tx) and (X, Ty) be two topological spaces. Let X-{a, b, c, d}. Tx-{4, X, {a}, {a, b}, {a, b, c}} Y…

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

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Q: 3A Given the Euclidean inner product space (R³(R),+,,*), where for each_x=(x1,X2,X3),…

A: The definition of inner product is given. We have to find the matrix representation of the inner…

Q: 1. Is V = why not? 1:0 20 and b≥( a vector space? Draw the graph and explain why or

A: By the answering guidelines we have to do first question is multiple questions are given so I am…

Q: 1. (a) Define what a linear subset of a projective space is.

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

Q: Every T3 space is regular space. True False

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

Q: (a) Define a reflexive space. Prove that a Banach space is reflexive if and only if its dual space…

A: To Prove :-  (a) A Banach space is reflexive if and only if its dual space is reflexive. (b) Closed…

Q: Prove that the extended Euclidean three-dimensional space ES3 is a three-dimensional projective…

A: Sol:- To prove that the extended Euclidean three-dimensional space ES3 is a three-dimensional…

Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: Prove that any linear map between finite dimensional normed linear spaces is continuous.

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Q: ive an example of a space is not vector space

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

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Q: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

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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: Q1) Prove or Disprove: Let L(F) be a linear space and A,B are two convex subsets of L. Then, AXB is…

A: The problem is related with convex set. Given that,                 LF  is a linear space  and A, B…

Q: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

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: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

A: We have to give an example of a topological space X such that the diagonal is not closed in X×X

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

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Q: Find and Verify two yuivalent Gpuiralent norms on sfrace R.

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Q: Give example on the inner product and same time is normed space

A: To find an example on the inner product and normed space. Definitions: Inner product space: Let V be…

Q: a) Find a space that is a regular but not Hausdorff. Justify your claim.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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

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Q: Differentiate between Euclidean and complex Euclidean spaces. How Schwartz inequality is modified in…

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Q: Every T2 (Hausdor ff) space is T1 space. True O False O

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Q: Justify this statement with example” in an infinite dimensional subspace need not to be closed”.

A: Infinite dimesional subspace need not be closed.

Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…