Show that the real number R with d(x,y)=|x-2y| are not a metric space

Q: 1- a = (1, .,0) in the usual metric space R. 2- (0,1) in the usual metric space R. 3- {(x, y): xy =…

A: As per our company guidelines we are supposed to answer ?️only first 3 sub-parts. Kindly repost…

Q: Use Hahn-Banach Theorem to show that a normed linear space X is finite dimensional if its dual X' is…

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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: 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: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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

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Q: Let (X,d) be metric space and X is unbounded, then d (z. y) . y EX, is 1+d(r,y) di de fined as dy =…

A: Metric space

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: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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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: 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: (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: = 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 (R. d) be discrete metric space, thend can not induces a norm on X O True O Fahe

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let X = R and def ine ||.|| on X as ||x|| = x² then X is normed linear space True False

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

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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: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

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

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

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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

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: Let X and Y be compact spaces. Then X × Y is compact.

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Q: let (2, m) be a meesurable space. then 1f f9: x LJare measural ble function, then -> ) f43 is mea…

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

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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: Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1…

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Q: metric

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Q: Let (X, t) is topological space, ACX, (X,d) is a metric space. Prove that: f: (X,d) -> (IR, Ts),…

A: Let ( X, d ) be a metric space and R with the standard topology. Then, we define a function  f(x) =…

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: 1. Show that if & is a function from a nonempty compact metric space X to itself such that d(@(x),…

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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: 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: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

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Q: (X,d) be a metric space. Define f: X x X -> Real Numbers by f(x,y) = d(x,y) / 1+d(x,y) . show that f…

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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: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…

Q: Let f: X, Y be a continuous surjection between metric spaces. If X is compact then A. X is complete…

A: Since X is compact , so it is complete. A option is true.