a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b. Give two examples of normed spaces that are not reflexive (with brief reasoning).

Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

A:

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: (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: Let X be a locally compact Hausdorf f space

A:

Q: 5. Prove that the property of a space having an isolated point is a topological property. 6. For any…

A: Sol

Q: Determine if the following statements are true or false. Briefly explain why. a) A quotient of a…

A:

Q: Prove the following: Suppose that {X; : i € I} and {Y; : i E I} are indexed families of topological…

A:

Q: Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c)…

A:

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: Let X and Y be normed spaces. Then show that the following hold : (i) If Y is a Banach space, then…

A: We consider an arbitrary Cauchy sequence (Tn)n≥1 in BL(X, Y) and show that (Tn)n≥1 converges to an…

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

A:

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

A:

Q: Construct examples to show that if X and Y are topological spaces with G acting on them such that…

A:

Q: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

A:

Q: If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff. The contrapositive of this…

A: Given : (1) (ℝ, T) with the topology induced by taxicab metric.              (2) (ℝ, T) with the…

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

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

A:

Q: Show that R is a bianch space.

A:

Q: The following statements about a topological sapce are equivalent: (a) X is a Hausdorff space; (b)…

A: Given: The statements: a) X is a Hausdorff space; b)  The diagonal D=x, x: x∈X is closed in X×X. To…

Q: let C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

A:

Q: Let X and Y be compact spaces. Then X × Y is compact.

A:

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

A:

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

A:

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

A:

Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

A:

Q: Q7: Prove that (R", ||-|| ) is complete metric space.

A:

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: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

A:

Q: Every T2 (Hausdorf f) space is T1 space. True O False O

A: Since T2 is a product preserving topological property. So T2 space is a T1 space.

Q: Let (R, d) be discrete metric space and A = N then A is complete and compact neither complete nor…

A: Given that (ℝ,d) is a discrete matric space and A=ℕ. then     d(x,y)=0 if x=y1  if x≠y A matric…

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

A:

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

A:

Q: The following statements about a topological sapce are equivalent: (a) X is a Hausdorff space; (b)…

A: We have to prove that the following statements are equivalent : (a) X is Hausdorff space (b) The…

Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step

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

A:

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: (2) Let (X,r)be a topological space, If for every A open set, and x E A", 3 Ux and Vy open sets,…

A: A topological space X is Hausdorff if for each pair of distinct points p and q in X , there are…

Q: (a) Let X and Y be normed spaces and F :X→Y be a continuous linear map. Show that F is closed . Is…

A: Given that,                   X and Y are normed spaces and F : X → Y is a continuous linear map. We…

Q: Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only…

A:

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

A:

Q: Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an…

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: 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…