61. Let Y be a subspace of X. A separation of Y is a pair of disjoint nonempty sets A and B whose union is Y, neither of which contains a limit point of the other. Then Space Y is connected if there is no Separation of Y

Q: If W is a subspace in an inner product space V and V = W + W, prove that V = WW¹.

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Q: Show that [0, ∞) and (0, 0) ad subspaces of R with the usual topology are not homeomorphic.

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Q: Let Y be a vector subspace of a normed space X. Show that the closure of Y is also a vector…

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Q: Show that a finite union of compact subspaces is compact.

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Q: 4. Show that the set S = {ax²+bx+c: a, b, c € R and a ‡ 0} is NOT a subspace of R[x] by showing that…

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Q: 1. Let A be a subset of normed space X. show that [4] is smallest closed subspace of X contain A

A: Statement:- Let Abe a subset of normed space X. Show that A is smallest closed subspace of X contain…

Q: 3A Given the Euclidean inner product space (R³(R),+,,*), where for each_x=(x1,X2,X3),…

A: The definition of inner product is given. We have to find the matrix representation of the inner…

Q: Show that [0,∞) and (0,∞) as subspaces of R with the usual topology are not homeomorphic.

A: Given: [0, ∞) and (0, ∞) are subspaces of ℝ. To show: [0, ∞) and (0, ∞)are not homeomorphic.

Q: 3. Consider the set C(R) = {continuous functions of real numbers and consider its subsets. Show that…

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Q: а. W- : x is a scalar |2x + y b. W = : x, y are scalars x + y c. W = +1 : x is a scalar

A: we have to check subspace.

Q: Please go through each option with explanation

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Q: 12. Show that the set of all continuous function on the interval [-1,1] is a vector space under…

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Q: If X, is a dense subspace of a normed linear space X show that X X'.

A: We will use basic knowledge of dual of normed linear spaces and the concept of isometric…

Q: (b) For subspaces X and Y, the sum X + Y is a direct sum (denoted X + Y) if every VEX + Y can be…

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Q: 6. Show that the closure Ỹ of a subspace Y of a normed space X is again a vector subspace.

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Q: Let Y be a closed subspace of a normed linear space X. Show that X is complete X are complete. Y AY…

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Q: 4. Let q: X→ Y be a quotient map. Suppose Y is connected, and, for each y € Y, the subspace q-¹({y})…

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Q: - If X is a vector space and if X= MN, where M and N are subspaces, prove that N is isomorphic to…

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Q: 1 = { x:= = < |x|<1} am an 1. Consider the set Y=[-1,1] as a subspace of R. Is A=< open set in Y? Is…

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Q: 3. This problem is about set addition and subspaces. Let d = and dz = and V = span{d}, %3D U =…

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Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

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Q: .: Remind that the set F of all functions from R to R is a vector space. Prove that the set E of…

A: We know that If W is a subset of a vector space V , then W is a subspace of V if and only if the…

Q: Show that a finite union of compact subspaces of a topological spaces X is compact-

A: We have to show that the finite union of two compact subspaces of a topological space X is compact.

Q: 4. Let H be the set of 2 × 2 whose trace is equal to zero. That is [a11 a12 Н- E M2(R) | a11 + a22 =…

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Q: 5. If W is subspace, if A = : a. (A + B) is subspace these [8] € W and B = Non of above [8] € W,…

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Q: II. Decide which of the following are subspaces of corresponding vector spaces: C R?

A: Vector space.

Q: 6. Let V be a vector space with subspaces U and W. Show that the set UnW = {x|x EU and xEW} is a…

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Q: Show that [0,1] and (0,1] as subspaces of R with the usual topology are not homeomorphic.

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Q: Prove that a closed linear subspace of a reflexive space is reflexive.

A: This is a problem of functional analysis.

Q: The set of all points in R3 satisfying x + y - z = 0 is a subspace. Note that the set of point…

A: It is known that any subspace of ℝ3 should contain an additivities identity element 0, 0, 0.