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

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: 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: Let r, y E R". Using the triangle inequality ||r + y|| < ||1|| + ||y||, show that r|| – ||y||| < ||r…

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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 €…

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

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Q: Let (R. d) be discrete metric space, then Ris separ able O True O False

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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: 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: The taxicab metric p on the plane R² is defined by the rule: p((x1, y1), (x2, Y2)) = |x1 – x2| + |y1…

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Q: メ=D¢ c omplex number %3D Doyou CGid) is metric space of not ??

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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: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

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

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

A: This is a problem of functional analysis.

Q: Let (X, d) be a metric space, and p: X→ IR doxy) I+desy) de fined by puM pusy) = %3D show that p is…

A: Consider a set X and a function defined from it to the real numbers as d : X×X→ℝ. Then, X,d is a…

Q: Find all limit points of a set A = {(2+,2 – |n e N} on the Euclidean plane with the isual metric. Is…

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

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Q: True or False: (1, i) and (-1, i) are orthogonal in the Euclidean inner product space C?. .

A: Consider the provided question,

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, 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: snip

A: Theorem:                  Suppose &lt;v,w&gt; is an inner product on a vector space V.  Then…

Q: 2.5 Show that any interval (a,b) inR with the discrete metric is locally compact but not 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 V be a real inner product space, and let u, v, w E V. If (u, v) = 1 and (v, w) = 3, what is (3u…

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Q: (b) Show that the metric space (Q,1) is not complete. (c) Prove that the discrete metric space (M,d)…

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

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

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Q: Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

A: Given:  (Y,d) is a metric space, and Y is totally bounded To Prove Y is bounded Definition: Let…

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

A: This is a problem of Functional Analysis.

Q: (b). Let V = R? = {(r,,1)| plication. Is it a vector space? IER} with usual addition and scalar…

A: Is it vector space

Q: Let a = (a, az); b = (b, ,b2) € R?, define the function %3D %3D d: R? x R? ---> R by d=(a, b) =…

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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: Use the Completion Theorem below to show that the Completion of the discrete metric space X is…

A: By completion theorem, there exists a complete metric space which has a subspace that is isometric…

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

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Q: What vector space property does Q2 ={(m, n) m, n ∈ Q} fail? *set Q: rational numbers

A: Q^2 is a vector space over Q but it is not a vector space over R.

Q: Let X,d) be a metrie space. Define d': Xx X >R by: d'Cxoy) Show that (X.d') is min (dl8), 1). a…

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Q: H.W.2 Exercise Prove that: 1. (IR, J- |) is a complete metric space. 2. (Q. |-1) is not complete…

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

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Q: Exercise. Prove that in any metric space (X, p), a closed ball {x E X : p(a, x) <r} is closed.

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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: (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: 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: (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: 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…