Let (X,d) be a metric space with the added condition that for any three points x,y,z in X, d(x,y) <= max{d(x,z),d(y,z)}. (a) Show that every triangle in X is isoceles. (b) An open ball in X with center u in X and radius r > 0 is defined as B(u;r) = {x in X | d(u,x) < r}. Show that every point in an open ball is a center for the open ball.

Q: 4. Suppose that (X, dx) and (Y, dy) are metric spaces, and define distance on X × Y by Prove that…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

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A: This is a problem of functional analysis.

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A: Given that X,d be a metric space and A⊆X be complete. The objective is to show that A is closed.

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

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A: here we only need to show all the condition of topology.

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A: Please check the answer in next step

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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: = 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.

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

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

A: This is a problem of functional analysis.

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: Let X be a normed space, Xn, Yn EX such that Xnx, Yny, then 1-Xn + Yn → 4- ||xn-ynll→→llx - yll.…

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Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

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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 finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

A: Sufficient to prove that M is closed and 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: Theorem 2: Let X be a finite dimensional normed space and let r > 0. Then, the closed unit ball B[0;…

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Q: If X is a metric space and Y C X, then Y is a metric space.

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Q: Let (R, d) be the metric space with d(x, y) = /[x – y[. Then the open ball O [-3, 5] ]-3, 5[ ]-1, 3[…

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Q: 14. If d is a metric on a vector space X‡ {0} which is obtained from a norm, and å is defined by…

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

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Q: 3 \et F be anormid space and- |2+j|l <llxll+yllprove that F has a stridtl (3) y convex- orm

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Q: If x is a normed linear space and the closed Unit ball M is compact, then X is finite dimensional.

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Q: Let X is a normed linear space and the ball M= {xe X:|| x ||≤1} is compact, then X is finite…

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