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

Q: (a) Define the T1-space.

A: Define the T1 - space

Q: 3. Prove that the connected components of a space X1,..., X, are sets of the form E x ...x En where…

A: Let Y be the connected components of a space X1X2...Xn. The connected components of X1X2...Xn are…

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

A:

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

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Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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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: 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: 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: Prove that the quotient of a locally connected topological space is locally connected.

A: We have to prove that Let q:X→Y be a quotient map, X is a locally connected space. Show that xY is…

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: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

A:

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

A: Sol

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: Prove that if X is a Banach space, X ∕= {0}, then there is a non-zero biounded linear functional on…

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Q: Let (X₁, || ||1), (X2, ||· ||2) be Banach spaces. On X₁ X X₂ define || (X1, X₂) || = || X1 ||1 + ||…

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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 that an affine map is continuous if and only if its linear part is continuous

A: If T:A→Rk is the affine map, where A is the affine span of an affine independent subset of Rn. Let…

Q: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

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

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

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

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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: 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: Show that any infinite subset A of a discrete topological space X is not compact.

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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: 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: / Let (X₁T) and (Y₁J) be top. Spaces and fix-x be a surjective and continuous map, then y is Compact…

A:

Q: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

A:

Q: Let (X, d) be a compact metric space and let A ⊆ C(X; C) = {f : X → C; f is continuous} be an…

A: please see the next step for solution

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

A: This is a problem of Functional Analysis.

Q: 6. Prove that if x1, x2,..., Xn are distinct points in a Hausdorff space X. Prove that there are…

A: Let X be a Hausdorff space. Then in X any two distinct points are separated by disjoint open sets.…

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

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Q: Let X be a topological space with the property that, for every topological space Y, every function f…

A: The space X with the discrete topology means every singleton of space is open.

Q: Examine the continuity of if 0 a at

A: We need to examine the continuity of the given function f(x) at x=a.

Q: Give an example of a norm space.

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Q: a discrete space

A: We know that in a discrete topological space X, every point in X is an open set.

Q: 3: Prove that any reflexive normed space is complete.

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

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

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Q: Finite topological spaces are always complete. True False

A: Finite topological spaces are always complete. So it is true statement.

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

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: (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: 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: Let X be an inner product space. Then show that the projection theorem holds if and only if Riesz…

A: Given that: X=Inner product space. The objective is to show that the projection theorem holds if and…

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: 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 X is Hausdorff if and only the diagonal A = {(x,x) | x € X} is closed in X x X.

A:

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.