Let Y be an infinite subset of a compact set X c R. Prove Y' # Ø.

Q: Prove that det(kA) = kn det(A) for any A ϵ Mnxn(F ).

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Q: Consider the reglon bounded by the graphs of y- and y- VE

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Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Then X" is homeomorphic…

A: Proof-) The Cantor set itself is a cantor space. But the canonical example of a cantor space is the…

Q: Let X be any set with the discrete topology. Show that X is normal.

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Q: (a) Show (by calculation) that {u1(x), u2(x)} is an orthonormal set. (b) Find the projection p(x) of…

A: Given the space C0,1 of continuous function on the interval 0≤x≤1, with the inner product defined as…

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: Prove (sn) is bounded if and only if lim sup |Sn| < +∞.

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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: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: Provethat amappi'ng from a metric SpaceXto metri'c space X is con tinuok ifand only f the inverse…

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Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: Show that a linear map T:X-→ Y between normed spaces is continuous if and only if {Tx,} is a Cauchy…

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Q: Let A ϵ Mnxn(F). Under what conditions is det( - A) = det(A)?

A: To Determine: Let A ϵ Mnxn(F). Under what conditions is det( - A) = det(A)? Given: we have a matrix…

Q: Show that R is not compact.

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Q: Functional theory and functional analysis

A: According to the given information it is required to prove that:

Q: 2/Define the topology If X-R² and T = {GK, KERZULAR} GK = {(XY) ER²|x+y >K3 Prove is topolgy on R²

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Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify tha the quotient…

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Q: O Let X and Y be normed spaces and F :X→Y be linear. Prove that F is continuous if and only if every…

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Q: Let X and Y be two topological spaces. X is connected and Y is disconnected are they homotopy…

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Q: Suppose (X,S,u) is a measure space and f,h: X → F are S-measurable Prove that I|fh||, < ||f||, ||4|a…

A: Given X,S,μ is a measure space and the functions f,h:X→F are S-measurable. Suppose 1p+1q=1r,…

Q: Let ( Xr d be ame tric space,show that exist atopology on X induce b

A: Let (X, d) be a Metric Space We know that in a Metric Space  (1) arbitrary union of open sets is…

Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient…

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Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: If S, S, are two subsets of an inner product space V(F) then show that

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Q: prove that the outer measure of Cantor Set C (x), 0<x< 1, is 1-x the generalized

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Q: Let T be a topology on X. Assume that T is Hausdorff and let x € X. (1) Show that {x} (and hence…

A: let T be a topology on x. T is a Hausdorff space

Q: Show that any infinite subset A of a discrete topological space X is not compact.

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Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

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Q: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

Q: 1) Prove that for any space, X, with any topology, T, that: (i) CI(X) = X and (ii) Int(X) = X.

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Q: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

Q: Prove that Cl(Q) = R in the standard topology on R. %3D

A: The closure of S may equivalently be defined as the union of S and its boundary, and also as the…

Q: Prove by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

A: In this question, we have to show that open balls in ℝn with the usual metric are not compact by the…

Q: Suppose the collection = {B₁ (x): xe (0,1)}. Is B an open cover of the interval [0, 1) CR? Give an…

A: Given: The collection B=B12x : x∈[0, 1). To determine: Whether B is an open covering of the interval…

Q: Let X and Y be normed linear spaces and F:X-Y. If for every Cauchy sequence {x, in X, the sequence…

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Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

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Q: Suppose U and v are conformally equivalent. Prove that if U is simply connected, then so is V.…

A: Given that U and V are conformally equivalent that is there exist a mapping f from U to V such that…

Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient…

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Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+…

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Q: Prove disprove that (R.Tco-finite) is Te-space or

A: To Prove - Prove or Disprove that ℝ, τco-finite  is T2 - space.       Definition used - T2 - space…

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: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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Q: Given, we have a finite purely non-atomic measure space (X, F, µ), prove that µ(F) = [0 µ(X)].

A: Given: Here, the given is finite purely non-atomic measure space (X,F,μ). To prove that μ(F)=0, μ(X)

Q: If V is finite dimensional ond f is a homomorphisni of V into itself which is not onto prove that…

A: If f is a homomorphism of V into itself, then f(0) = 0 .

Q: Let X and Y be normed linear spaces and F : X → Y be linear. Prove that F is continuous if and only…

A: Let X and Y be matric space and let f: x→y be a uniformly continuous function. If ( xn)n∈N is a…

Q: 1. Let X C R"-1 be bounded. Prove that X considered as a subset of R" is a volume zero set.

A: Given, X⊆ℝn-1 That is, X is a subset of ℝ in terms of exponential n. So, let we consider the set,…

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.