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

Q: Use Hahn-Banach Theorem to show that a normed linear space X is finite dimensional if its dual X' is…

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: 4) Show that the subspaces of R? are precisely {0}, R2, and all lines in R? through the origin.

A: A subspace of a vector space is itself a vector space that is contained in another vector space.

Q: if a family iP; Sierof sominorms on avectorspace X separates distinct points of x , show that the…

A:

Q: (2) Find all connected covering spaces of RP2 v RP².

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Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: 6. Show that a compact metric space X is locally compact.

A: The given problem is related with metric space. We have to show that a compact metric space X is…

Q: 4. Determine whether M22 = { 1: a, be R} : a, be R M22 is not a vector space

A: Check vector addition

Q: Prove that any Space is complete. finite dimensional normed

A: Introduction: Completeness is a property of a topological space. When in a topological space, every…

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: * Every Pre-Hilbert space is I - Continues Normed space -orthogonal Metric space inner Produced

A: Every pre-Hilbert Space is a complex vector space X together with an inner product (A complex inner…

Q: . Prove or disprove every metric space is normed space,

A: A normed vector space or normed space is a vector space over the real or complex numbers, on which a…

Q: If X is vector space and (X,d) is discrete metric space, then X can induce an inner product space…

A:

Q: 1. (a) Define what a linear subset of a projective space is.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

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: 4. Show that equivalent norms on a vector space X induce the same topology for X.

A:

Q: Any finite dimension vector space with dim(X) is equal to n is isomorphism with F dows but O True O…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Every finite dimensional vector space is

A: We have to prove the given statement.

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

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Q: Every finite dimensional vector space is algebraically reflexive. Prove that.

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Q: Show that a subspace of a Hausdorff space is Hausdorff.

A: Prove that, a subspace of a Hausdorff space is Hausdorff.

Q: If A is 3 × 3 with rank A = 2, show that the dimension of the null space of A is 1.

A: Given: A is 3*3 matrix with rank A = 2

Q: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

A:

Q: 2. Show that if the orthogonal dimension of a Hilbert space H is finite, it equals the dimension of…

A: Given: Finite dimension To find: Former finite

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

A:

Q: Given any nontrivial Euclidean space sis (e₁,..., en) for E we can construct an ortho perty that for…

A:

Q: Prove the locally compact normal linear space is finite dimensions.

A: We have to prove the  locally compact normal linear space is finite dimensions First we define the…

Q: that set of of degree subspace of P₂ Prove or disprove all Spolynomials is 43 not? Or

A: P2 is vector space of real polynomial of degree less or equal to 2. That is P2=px=a0+a1x+a2x2:…

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

A: This is a problem of functional analysis.

Q: prove that any finite dimensional normed space is reflexive.

A: Let us suppose that we have a finite-dimensional normed space, X For any natural number, n, let…

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: Prove that a finite dimensional normed linear space is always reflexive.

A:

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

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Q: 12) Show that subset W of a vector space V is a subspace of V if and only if span(W) = W.

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Q: Let N and N' be normed linear spaces with the

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

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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.

A:

Q: If V1, V2, V3, V4, and W are all vector spaces.  Are L(V1x...xV4, W) and L(V1, W) x...x L(V4, W)…

A: Given that: V1,V2,V3,V4,W=Vector space The objective is to verify the given statement. Using the…

Q: Theorem 9.8. A metric space is Hausdorff, regular, and normal.

A: The objective is to show that every metric is hausdorff, regular and normal. Let x≠y be points of a…

Q: SHOW THAT THE SET OF POLYNOMIUMS HAS A VECTOR SPACE OF POLYNOMIALS OF A LESS DEGREE THAN “g”  (Pg…

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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: 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: SHOW THAT THE SET OF POLYNOMIUMS HAS A VECTOR SPACE OF POLYNOMIALS OF A LESS DEGREE THAN “g”  (Pg…

A:

Q: (4) $5.11, Suppose X is a subspace of an inner-product space V. Show that X is a subpace of V.

A: