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

Q: Prove that any contraction between metric spaces has at most one fixed point

A: According to the given information, it is required to prove that any contraction between metric…

Q: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

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: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

Q: show that the discrete metric on u Vector space X+3 of eannot be obtained from a norm.

A: To Show: Show that the discrete metric on a vector space X ≠ 0 cannot be obtained from a norm.

Q: If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensional

A: This is the question of Functional Analysis.

Q: Let (R, d) be usual metric space, then Ris ot separable True False

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Q: If X =R and d(x.y)30, then X metric space not metric semi metric

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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: The Cartesian product X = X1 x X2 of two metric spaces (X1, d1) and (X2, d2) can be made into metric…

A: 4) To define a new metric on X which is different from the metrics mentioned in (1), (2), and (3).…

Q: Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded

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Q: (a) Suppose that (M, d) is a metric space. Consider the mapping f: M [0, 1] be given and let di(x,…

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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: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded. %3D

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Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + 3y2 = 12. Show that H is a closed subset…

A: Please check the answer in next step

Q: Prove that if f is a continuous mapping of a compact metric space, X, into a metric space Y, then…

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Q: A subset I of a metric space R with the usual metric is compact if and if only it is an interval…

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Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: 2. If X,Y are Banach spaces on XOY define |(x, y) = |, +|yly . Show that this gives a norm making X…

A: Given: Dual of c0  is isometric to l1 State the principle of uniform boundedness.

Q: 3. Show that X = [0; 1] with the discrete metric is bounded but not totally bounded

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Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a…

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Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…

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Q: (v) Prove that if the space (X,T) has the fixed-point property and (Y, T1) is a space homeomorphic…

A: Given: v) To prove the given statement as follows,

Q: If X is vector space and (X,d) is discrete metric space, then X can induce an inner product space…

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Q: (K", || |,) is called the LP-norm on K". is a normed linear space for 1 <PS00. || |,

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Q: A. Let H be the set of all points (x, y) in R? such that x2 + 3y2 = 12. Show that H is a closed…

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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: Let (x, g) be a metric space. Is the function d (x, y) = k.g (x, y) (k € R) defined from XxX to R a…

A: Let X,g be a metric space . Given that function X×X to R defined as  dx,y=k·gx,y where k∈R A metric…

Q: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

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Q: Let X be the subset (R x {0}) U (R x {0}) CR2 Show that the quotient space X/~ is locally Eue line…

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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: Show that every closed ball in R" is an n-dimensional manifold with Assuming the theorem on the…

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

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Q: 0:1 Show that norm ||x|| = (Elx¡l?)ž satisfies the conditions of normed space. %3D

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Q: Q7: Prove that ( IR", ||·|| ) is complete metric space.

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Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

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Q: Let (X,d) be a metric space with the added condition that for any three points x,y,z in X, d(x,y) 0…

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Q: Show that the closed ball Y in a metric space (x,d) is a closed set .Also show that if (x,d) is…

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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: Give the definition of a separable metric space (S, d). Show that the metric space (R, d), where…

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

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Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

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Q: (b) Prove that the unit ball of a normed linear space is compact if and only if the normed linear…

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Q: The set R2 is a vector space over R. How Prove it ?

A: We need to check each and every axiom of a vector space to know that it is in fact a vector…

Q: Prove the parallelogram law on an inner product space V; that is, showthatllx + Yll2 + llx - Yll2 =…

A: Prove the parallelogram law on an inner product space V; that is,  Show that…

Q: (a) Prove that the metric space (Z,) is complete.

A: Complete Metric Spaces: A metric space (X,d) is said to be complete iff every Cauchy sequence in X…

Q: Give the definition of a separable metric space (S, d). Show that the metric space (IR, d), where…

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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…