1. Let X be a metric space and A C X be a metric subspace. (a) If A is closed in X, prove or disprove that A is complete. (b) If A is complete, prove or disprove that A is closed in X.

Q: 4. Check that the following are vector spaces: (a) RN. 55 / 600 (b) CN. (c) C(N), where 2 is a…

A: A non-empty set V, +, · is said to be a vector space over a field F together with operations of…

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: The subset of pairs {(x, y) E R²|x,Y are integers } in the vector space R².

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Q: 4. Decide if the following statement is True or False: Let U, W, and X be subspaces of a finite…

A: Given - Let U, W and X be subspace of a finite dimensional vector space V such that U ∩ W = 0 and U…

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

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Q: Please go through each option with explanation

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Q: A shopper buys cat food in bags of 3 lbs. Her cat eats 3434 lb each week. How many weeks does one…

A: Given that A shopper buys cat food in bags of 3 lbs. Her cat eats 3/4 lb each week

Q: . Exercise 1.2.10 Let V₁ and V₂ be normed spaces with norms || ||1 and ||||2. Recall that the…

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Q: If V is a 7-dimensional vector space and S is a set of 10 vectors, then the elements of S must be…

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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: Let V be a 8 dimensional vector space over the set of reals. Then the dimension of a subspace of V…

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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 X be a vector space. Let || and || || be two norms on X. When are these norms said to be…

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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: A solid concrete porch consists of 3 steps and a landing. The steps have a tread of 11 inches, a…

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Q: i. Let (B, || – || B) be a Banach space and let A C B be a subspace (as a vector space) of B. Recall…

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Q: (a) Let F be a finite dimensional subspace of an inner product space X. Then show that X=F+ F-and F…

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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: Explain why the following statement is false:                          “A subspace is any subset of…

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Q: . In the following either prove that the set W is a subspace of the vector space V or give a…

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Q: 18. Let V= {(a1, a2): a1, a2 € R}. For (a₁, a2), (b1,b2) EV and c € R, define (a1, a2) + (b₁,b2) =…

A: An operation vector addition ‘ + ‘ must satisfy the following conditions:Closure : If x and y are…

Q: (a) Prove that the metric space (2,11) 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 any linear map between finite dimensional normed linear spaces is continuous.

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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: f X is an arbitrary topological space, then we have the following: (a) each point in X

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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: 1. Let X be a topological space. Suppose X contains an infinite, closed, discrete subspace. Show…

A: Given topological space X such that X contains an infinite closed discrete subspace. We now show…

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: Prove that a linear map on a normed vector space is bounded if and only if it is continuous.

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Q: (b) Suppose -|, and ||-||2 are two equivalen norms on a vector space X. If (X, ||.||,) is complete,…

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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: 3. Determine, with proof, whether each set is a subspace of the given vector space {a, + a1x + a2x²…

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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 metric subspace ]1,2] of the Euclidean metric space R is a complete metric space. False True

A: Solution

Q: 4. Let W be a subspace of a vector space V. We define a relation: V₁ V₂ if V₁ V₂ € W. ~ 4a. Show…

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Q: 13- Suppose that W is a subspace of normed space V then it is not necessary that W is also normed…

A: Subspace of a normed space