Prove that if f is a continuous mapping of a compact metric space, X, into a metric space Y, then f(x) is compact.

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

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

Q: Prove that if f: X → Y is a continuous open mapping, X is compact, and Y is connected and Hausdorff,…

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Q: Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norn of function f(x)…

A: This is a problem of functional analysis.

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

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Q: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

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Q: Exercise 2. Let X = C[0, 1], where C[0, 1] is the set of continuous functions on [0, 1], and the…

A: A set (X,d) is called a metric space if these four properties are satisfied for all x, y, z ∈ X: d…

Q: Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric…

A: A matric space X is separable if it has a countable dense subset.

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: 2) let X and y be metric spaces and T. XY Y a ContinUOus mapping Then the image of a Compact subset…

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Q: Let X be a locally compact Hausdorf f space

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Q: A completely regular space is also T1-space 1 True 2 False

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Q: (a) Define a metric space (X, p). (b) Let p: R+ → R+ be a function defined by ln (2). Prove that p…

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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: . Prove or disprove every metric space is normed space,

A: A normed vector space or normed space is a vector space over the real or complex numbers, on which a…

Q: Let X be a disconnected space, that is X = UUV, where U and V are nonempty open subsets of X and…

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Q: (a) Prove that every finite subset of a metric space is closed and has umulation points.

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

Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: If S is a closed bounded subset of a metric space X, then S is compact.

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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: Show that ℓ^1 is a normed linear space.

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Q: Let f be a continuous mapping of a metric space X into metric space Y,g be a continuous mapping of…

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Q: Show that R is a bianch space.

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Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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Q: Assume that (M, d) is a compact metric space. Show that if f: (M,d) → (Y, d) is continuous and…

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Q: Let X= {A ⊆ ℝ| 1 is not in A and ℝ \ A is finite}. Prove that ℝ is a compact Hausdorff space with…

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Q: Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

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Q: let X be a firite dimensional notm Space Then Prove that is compat.

A: As X is a finite dimensional norm space, this means that, dimX is finite.

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

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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 (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: A of a metric space (x, d.)

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

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Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

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: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: Show that it A is subset of X any with the discre te metric then %3D

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Q: Let (X, d) be a metric space, f,g:X → R some functions and xo E X, q E R. Assume that f(x) = g(x)…

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Q: Let X and Y be metric spaces. Assume that E is a dense subset of X, and f, g: X Y are continuons…

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Q: Every inner product space is a metric space O True O False

A: TRUE

Q: Let X be a compact subset of R and let C(X) denote the set of continuous real-valued functions on X…

A: let X be a compact subset of ℝ and let CX denote the set of continuous real valued functions on X.…

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: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: Let E,F ⊆ X,d be compact subsets of a metric space X. Just using the definition of compactness…

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

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Q: Let (X, d) be a compact metric space and T: XX be a mapping that satisfies d(Tr,Ty)<d(r,y) for all…

A: What is Metric Space: Metric space is an abstract set in mathematics, particularly topology, with a…

Q: Give an example of a continuous function T, and an open set O in a metric space (X, d) such that…

A: Consider the provided question, Hello. Since your question has multiple parts, we will solve first…

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: Let E and F be compact sets in a metric space (X,d). Show that E U F is compact using the definition…

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Q: Let (X, || · ||) be a normed linear space. Prove that (a) B1(0) = B1(0).

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