Show that in the finite complement topology on R every subspace is compact. Is R compact in the lower limit topology?

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: Suppose U, W,, W, are subspaces of a vector space V. Show that (Un W,) + (UnW,)Cun (W, + W,) Find…

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Q: Let M, N be closed linear subspaces of a Hilbert space H. If M 1 N, prove that M + N is closed.

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Q: 3. Show that closed subspace of a complete metric space is complete. 4. Let f be a continuous…

A: As per norms, the first question is answered: To prove that a closed subspace Y of a complete metric…

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: Let H be a subspace of real space R. In terms of a geometrical definition of H, which of the…

A: Given question :- Let H be a subspace of real space R3. In terms of geometrical definition of H,…

Q: Let T and T' be topologies on a set X such that T' is strictly finer than T. If Y is a subset of X,…

A: Definition:  Let Τ and Τ' be topologies on a set X. If Τ' is strictly finer than Τ, then Τ⊂Τ'.…

Q: Let W be a finite-dimensional subspace of an inner product space V. Show that if T is the orthogonal…

A: Given:W be a finite-dimensional subspace of an inner product space VTo prove:If T is the orthogonal…

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: Construct the vector space of all 3-tupies over GF(3). Form a two-dimensional subspace and its dual…

A: Given, the vector space of all 3-tuples over GF(3).

Q: Find an example of a unitary operator U on an inner product space and a U-invariant subspace W such…

A: Let V be an infinite dimensional inner product space consisting of finite sequences.Define U:

Q: can a subspace of R^n have a dimension less than n.

A: Yes , it can be .

Q: If V is a finite dimensional vector space and W is a subspace, the W is finite dimensional. Prove it

A: NOTE:  If V is a finite dimensional vector space and W is a subspace of V, then W is finite…

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 V and W be subspaces of R", and dim V and dim W are strictly less than n. Assume that no…

A: Given:- V and W are subspaces of ℝn              dim V<n, dim W<n              Also, there is…

Q: The empty subset of any vector space V is not a subspace of V. True False

A: The empty subset of any vector space V is not a subspace of V The given statement is false.

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: Let M be a closed linear subspace of a Banach space X. Prove that the quotient space X/M, with the…

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Q: show that if w is a subspace of a finite - dimensional vector space V and dim (w)= dim (V) then W =…

A: In this question, we will use the fact that If dim(W) = dim(V ) = n, then a basis for W is a…

Q: Let V be a finite-dimensional inner product space, let W be a subspace of V, and let projw : V → V…

A: Please check step 2 for the solution.

Q: Let W be a subspace spanned by the u's, an

A: Given, W be a subspace spanned by u's, and write y as the sum of a vector in W and a vector…

Q: Let X and Y be a topological spaces and assume that A C X and B CY. Then the topology on Ax B as a…

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Q: 2. Provide an example of a subset U of V that is a subspace of V, as well as an example of subset W…

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Q: Any nonzero subspace W C R" has an orthonormal basis. True O False

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Q: Prove that the dual of l° is not isometriç to, 1', but contains a proper subspace that is isometric…

A: Consider that there are bounded functions on l∞ which are not of the form fy=∑xkyk for some x. Let,…

Q: Suppose V is an interior multiplication space with finite dimension and W, and W2 are subspaces of…

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Q: Find the orthogonal projection of onto the subspace W of R' spanned by -1 -1 -1 1 1 projw (7)

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Q: Prove that Co is a Banach space by showing that it is a closed subspace of l

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Q: Prove that the dual of 1 is not isometric to 1¹, but contains a proper subspace that is isometric to…

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Q: Prove that X is completely regular space if and only if it is homeomorphic to a subspace of a…

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Q: Let X and Y be a topological spaces and assume that A CX and B CY. Then the topology on A x B as a…

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Q: Let T and T' be topologies on a set X such that T' is strictly finer than T. If Y is a subset of X,…

A: Please comment if you need any clarification. If you find my answer useful please put thumbs…

Q: (b) Prove that if V is finite dimensional, then every subspace of V is also finite dimensional.…

A: (b)

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

A: Solution

Q: Verify that if W is a subspace of an inner product space Vand v is in V, then perpw(v) is orthogonal…

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Q: If M is linear subspace of hilbert space H then M is clos

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Q: If M is a closed subspace of a Hilbert space H then H %3DМӨМ'.Prove.

A: Complete inner product space is called Hilbert space

Q: Show that the set of all polynomials of degree at most 3 forms a subspace of Pn.

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Q: = E In > 1} be viewed as a subspace of R. What is the subspace topology on X? A. Discrete B.…

A: Let, y=1n: n≥1 be viewed as a subspace of ℝ.

Q: a) Prove that an operator P is the pr osed linear subspace M of a Hilbert

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Q: Let W be the subspace spanned by Note that this basis is not orthogon Tind the orthogonal projection…

A: We have to find the orthogonal projection of into W.

Q: Let X be complete metric space, then a subspace M of X is itself complete iff the set M is closed in…

A: Hint: First suppose that M is compete then show that M is closed then suppose that M is closed and…

Q: Let A and B be compact subspaces of a Hausdorff space X. Prove that AUB and AnB are compact.

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Q: Let X be a normed linear space which is non-zero. Prove that X is a Banach space if and only if the…

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Q: If M‡ Ø is any subset of a Hilbert space H, show that M is the smallest closed subspace of H which…

A: Solution:Hilbert space H, M⊥⊥ smallest closed subspaceWe first show that for any M⊂H, M⊥is closed.…

Q: Show that if W is a subspace of a finite-dimensional vec- tor space V, then W is finite-dimensional…

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Q: Suppose V is a finite-dimensional inner product space Suppose U, W are subspaces of V . Prove that:…

A: Given: V is a finite dimensional inner product space and U,W are subspace of V. To prove :…

Q: Let H be a subspace of real space R. In terms of a geometrical definition of H, which of the…

A: Given: