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

Q: Use the sub no spaces: T-2y = - %3D 3x + 2y

A: Given that: x-2y=-10           ........(1)3x+2y=-14      ..............(2)

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: Бxї Zet (x, г) бе space, then Co-countable тебя xis not Тз-брасе

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Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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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 = (1, 2, 3), Tx = (0, X, (1), (2, 3}}, then (X, Tx) is %3D %3D 1) a T1 space 2) a T2 space

A: T1 space: A topological space X has the T1 property if x and y are distinct points of X, there…

Q: (b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ?…

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Q: Let V be a real inner product space, and let u, v, w E V. If (u, v) = 1/2 and (v, w) = -2, what is…

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Q: If X is an inner product space and A = {0} , then A- = {0}O AO

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Q: Let (X, d) be a metric space and let A ⊆ X be complete. Show that A is closed.

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

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: Let X be a locally compact Hausdorf f space

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Q: A completely regular space is also T1-space 1 True 2 False

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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: 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: 5. Let (X,r) and (Y,A) be two topological spaces. Prove that a function f:(X.r)(Y.A) is closed if…

A: Since you have posted multiple question according to company rule we are supposed to give solution…

Q: Every T3 space is regular space. True False

A: Since a regular T1 space is called a T3 space.

Q: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

A: Given, X and Y are normed spaces and X ≠ 0. We have to prove that BLX , Y is a Banach space with the…

Q: Let X and Y be topological spaces and fix some yo E Y. Show that the map f: X → X x Y defined by f…

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Q: If X is an inner product space and A, BC X, then AIB iff An B = {0} True O False O

A: Given that,

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

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: Let X ų be topological spaces and f: X→Y be a continuous map. If X is completely normal then Y is…

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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 R be the set of real numbers and d: R x R R be defined as follows. -> 2|x-y d(x, y) = 3+|x-yl…

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Q: Let (X, T1) and (Y, T2) be topological spaces. Show that the set {U × Y ∣ U ∈ T1} ∪ {X × V ∣ V ∈…

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Q: Find the dimensions of the following linear spaces. (a) R2x3 (b) The real linear space C6 (c) Ps

A: A matrix is given R2×3, It means this matrix have, No. of rows = 2, No. of column = 3.

Q: Let [a, b] be an inter a space. Let ƒ:[a,b] → X

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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: 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: * Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((r, y)) = x…

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Q: c spaces. Show that X x Y and Y

A: The sup-metric d1 on X×Y is defined as d1x1, y1, x2, y2=max dx1, x2, ey1, y2. Similarly, the…

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: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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Q: Let X=R and T = {]a, +oo[/a R}U{o,R}, then (X, 7) is a regular space. True False

A: This is a problem of topology.

Q: be a n-dimensional normed space, then prove that its dual space is also n-dimensional.

A: Let X be a normed linear space. Let X* be its dual space with the usual dual norm ||T|| = sup{…

Q: Let X be a normed linear space such that its dual X' is separable, then show that X is separable ?…

A: This is a problem of Functional Analysis.

Q: (a) Let X and Y be normed spaces and F :X→Y be a continuous linear map. Show that F is closed . Is…

A: Given that,                   X and Y are normed spaces and F : X → Y is a continuous linear map. We…

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

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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: Let X and Y be metric spaces and f : X Y be continuous. Let S C X. now: (a) f(S) C f(S). (b) If S is…

A: Let X and Y be metric spaces.

Q: Find isomorphic spaces O R and R² O P3 and M2x1 O P3 and M2×3 O P3 and R2 O R' and M2×2

A: We have to choose whether the given spaces are isomorphic or not.

Q: if x is a compact Metric space and F e C (X₂Y) is equicontinuous, then F i's uniformly…

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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: 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: Which of the following vector spaces is not isomorphic to all the others R6  R6[x]  M2×3(R)…

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

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Q: Let X and Y be two discrete spaces, then *

A: Note: Since you have posted multiple questions we are only allowed to solve the first one. Thankyou…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.