Prove that the convex hull of a bounded set is bounded.

Q: Show that every second countable (A2-space) space is separable.

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Q: (a) Define the T1-space.

A: Define the T1 - space

Q: if a family iP; Sierof sominorms on avectorspace X separates distinct points of x , show that the…

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

A: This is a problem of Topology.

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: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

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Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

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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: Find the transitive closure of R={(0,1),(0,2),(1,1), (1,3),(2,2),(3,0)}

A: Given relation is                              R=(0, 1), (0, 2), (1, 1), (1, 3), (2, 2), (3, 0) we…

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: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

Q: Exercise 4.2. Let X be a space with the finite complement topology. Show that X is T1.

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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: 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: If X is a normed space then the open ball and closed ball is a Example: convex set

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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: Each nonempty can be represented as the convex hull of its extreme points. --------

A: Each of the nonempty compact convex set can be represented as the convex hull of its extreme point.

Q: Generate the valid inequalities needed to describe the convex hull of the following sets S. • S:=…

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Q: Generate the valid inequalities needed to describe the convex hull of the following sets  S = {x,y}…

A: The set S is S=x,y:x+y>b,x>0 integer,y>0. Let a=x1,y1∈S and b=x2,y2∈S. Since a=x1,y1∈S,…

Q: Exercise 2.27. Show that a set U is open in a topological space X if and only if every point of U is…

A: Let, U is an open set in a topological space X. Then, U⊂U  So, U⊂int(U) and int(U)⊂U Hence, U=int(U)

Q: Prove that a closed convex set in R" has a finite number of extreme points if and only it is the…

A: A closed convex set in ℝn has a finite number of extreme points if and only if it is the…

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: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

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

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Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

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: A finite set in a metric space cannot have accumulation points.

A: We have to prove that A finite set in a metric space can not have accumulation (Limit) point.

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

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Q: Show that any compact set in R is closed and bounded.

A: Proof

Q: Can you show an example of Null space?

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Q: Find the transitive closure of R if M. is

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Q: Find the transitive closure of R if MR is

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Q: Prove that the open set (0,2) is not a compact subset of ℝ by finding an open cover with no finite…

A: To prove:  (0, 2) is not a compact set of ℝ.

Q: Prove the locally compact normal linear space is finite dimensions.

A: We have to prove the  locally compact normal linear space is finite dimensions First we define the…

Q: What is the Boundedness Theorem?

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Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

A: Compact Set: A collection A of subsets of space X is said to be cover X or covering of X if the…

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

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

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Q: Show that all open balls are bounded subsets of a metric space.

A: Solution: Let M=X,d be a metric space. For an arbitrary point a∈X, the open ball Bεa where ε>0 is…

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

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Q: Describe in your own words the convex hull of a set of points in S in the plane. Show that the…

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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: Generate the valid inequalities needed to describe the convex hull of the following sets S. • S:=…

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Q: Sketch the graph of the convex hull of S. In R, let S =. :0sx<3

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

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

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.