3. Show that closed subspace of a complete metric space is complete. 4. Let f be a continuous mapping of a compact metric space X to metric space Y. Prove that f (X) is compact. A 1S

Q: iii) True or False: Every subspace of a complete metric space is complete.

A:

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

A:

Q: Let X₁ be a closed subspace and X₂ be a finite dimensional subspace of a normed space X. Prove that…

A: Here we have given that X1 be a closed subspace of a normed space X. So, the quotient space XX1 is…

Q: ) Suppose TE L(V). Prove the the intersection of every collection of subspaces of V invariant under…

A:

Q: Suppose X is a topological space whose topology is coherent with a family B of subspaces.  Prove…

A: Given X is a topological space, whose topology is coherent with a family B of subspace.Using the…

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

A:

Q: If X is the subspace of `∞ consisting of all sequences of zeros and ones, what is the induced metric…

A: We have given that ,  X is the subspace of l∞ consisting of all sequences of zeros and ones. We need…

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: Q2\ A- every finite set of a metric space is compact? B- S' US is the smallest closed set in M such…

A: Note :Our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: Let V be a vector space over an infinite field. Prove that V is not theunion of finitely many proper…

A:

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

A:

Q: Let V be an inner product space, and let W be a finite-dimensional subspace of V. If x $ W, prove…

A:

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

A:

Q: Show that in the finite complement topology on R every subspace is compact. Is R compact in the…

A: Recall that, a set  is compact if and only if every open cover has a finite subcover.

Q: Show that if Z subspace of a space Y and Y is a subspace of a space X then Z is a subspace of a…

A:

Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.…

A:

Q: If X a normed space and y is a proper proper Linear subspace of X. show that y has empty interior.

A: We proved by contradiction. Suppose Y has a nonempty interior then Y contains a open ball. Then we…

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: Let V be finite dimensional vector space and let W c V be subspace with complimentary subspace W'.…

A:

Q: Let E, be a closed subspace and E, be a finite dimensional subspace of a normed subspace X. Prove…

A: This is a question of Functional Analysis.

Q: If V is infinite-dimensional and S-is an infinite-dimensional subspace, must the dimension of V/S be…

A:

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

A:

Q: Show that a subset W of a vector space V is a subspace of V if and only if span[W] = W.

A:

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. Every complete metric subspace of a metric space is closed. a. True b. False

A: The solution is given below:

Q: Suppose that W1, W2,. , Wm and V are all finite dimensional vector spaces over F. Prove that 2,...…

A:

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

A:

Q: Show that if Z subspace of a space Y and Y is a subspace of a space X then Z is a subspace of a…

A:

Q: Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W.

A: To show that W is a subspace of V if and only if span (W) is equal to W.Let W be the subspace of V.

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

A:

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

A: This is a problem of functional analysis.

Q: Let A C X. Prove that if d is a metric for the topology on X, then d[(A × A) is a metric for the…

A:

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: 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

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

A:

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: Prove that the dual space of 10 is not isometric to 11 but contains a subspace isometric to l1.

A:

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: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

Q: (a) If E, and E, are open sets in a normed space X, then E, + E2 is open in X. (b) Every continuous…

A: Given:  a) If E1 and E2 are open sets in a normed space X, then E1 + E2 is open in X. b) Every…

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

A:

Q: Let Y be a subspace of a Banach space X, and let the norm on Y be the restriction of the norm on X…

A:

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

A:

Q: 8. If Z is an (n−1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null…

A: Given: n-1 dimensional subspace To find: Scalar multiple

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: x=y=z

A:

Q: Suppose V is finite-dimensional. Prove that every linear map on a subspace of V can be extended to a…

A: