) Let X be a finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

Q: Let X be a normed space and aɛ X. Show that there exists an fe X' such that ||a|| = Sup { \f(a)| :…

A: Given that,                X is a normed space and a ∈ X . We have to show that there exists an f ∈…

Q: If τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normal space…

A: τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normalspace or…

Q: 1. Prove that for every non-gero normed space X there is a non-zero bounded linear fmetional acting…

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Q: (2.3.15. Let X be a metric space. (a) Prove that SCX is bounded if and only if diam(S) < 0o.

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

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

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Q: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

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Q: Let E1, E2, ..., En ⊂ X,d be compact subsets of a metric space X. Just using the definition of…

A: For a metric space we know that the for a compact set every open cover has a finite subcover. So…

Q: Let V be an inner product space, and suppose that T: V→V is linear and that ||T(x)|| = ||x|| for all…

A: Given:

Q: If X is an inner product space and A = {0} , then A- = {0}O AO

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Q: Q3 Prove : If V is a normed linear space, and WcV, and U a closed subset of W Then given ve V,3u"…

A: given if V is a normed linear space, W⊂V and U is a closed subset of W then for v∈V there exist u*∈V…

Q: Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded

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Q: 3.3.9. Given a normed space X, prove that the function p: X R defined by p(x) = ||x|| is continuous…

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Q: Let D be a Euclidean domain with measure d. Prove that u is a unitin D if and only if d(u) = d(1).

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Q: Let X be a normed space. (a) If E₁ is open in X and E₂ C X, then E₁ + E₂ is open in X. Proof: (a)…

A: It is given that X be a normed space, E1 is open in X and E2 be any subset of X. We have to explain…

Q: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

A: Given, X and Y are normed spaces and X ≠ 0. We have to prove that BLX , Y is a Banach space with the…

Q: Let X and Y be normed linear spaces, and T E B(X,Y). Prove that 38 > 0 such that 8||x|| < ||T(æ)||,…

A: Consider the given information.

Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: Prove that: 1. (R, |. |) is a complete metric space. 2. (Q.I-1) is not complete metric space.

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Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

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Q: Let X= {A ⊆ ℝ| 1 is not in A and ℝ \ A is finite}. Prove that ℝ is a compact Hausdorff space with…

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Q: If S be a n-dimen Sional normed Space, then prove that its dual space is al so n- dimen Sional .

A: A normed space E is called a Banach space if it is complete, i.e., if every Cauchy sequence is…

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: Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

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Q: Find a norm for C([0, 1]), so that C([0, 1]) is not complete under this norm.

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Q: Is the following statement True or False? Justify each answer. If S is a closed bounded subset of a…

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Q: 0:1 Show that norm ||x|| = (Elx¡l?)ž satisfies the conditions of normed space. %3D

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Q: Let K be a compact subset of a metric space X, and assume that -R is continuous. Prove that f…

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Q: (a) Prove that every closed subset S of a compact metric space (M,d) is compact in M.

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

Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

Q: let a-Rbe ametric sPacesConsir he following subsed ofP @ Show that Dra De are comPact Setsornot of D…

A: Question from real analysis.

Q: Theorem 2: Let X be a finite dimensional normed space and let r > 0. Then, the closed unit ball B[0;…

A: Theorems and definitions we are going to use: 1. Definition: (X,|| . ||) is said to be bounded if…

Q: If the unit sphere {xe X : x = 1} in a normed space X is complete, prove that x is complete.

A: Given that X is a normed space. Let S1x=x∈X| x=1 be unit sphere of the normed space X. Given that…

Q: Q3) Let L = Z* and |x ||= max {]x1 - x3l, x2l, 15x,1} Vx = (x1, X2, X3, X4) E Z*. Is (Z*, ) is a…

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Q: H.W.2 Exercise Prove that: 1. (IR, J- |) is a complete metric space. 2. (Q. |-1) is not complete…

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Q: Every inner product space is a metric space O True O False

A: TRUE

Q: 10. If A is 3 x 3 with rank A 2, show that the dimension of the null space of A is 1.

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Q: Q6: (a) In any metric space (S, d), prove that every compact subset T is complete. (b) Prove that…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

Q: if x is a compact Metric space and F e C (X₂Y) is equicontinuous, then F i's uniformly…

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Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…

Q: An Elliptical norm for R2 is defined by + 4u2, where x = (1,u). (i) Prove that l is a norm for R2…

A: This is a problem of Functional Analysis.

Q: (a) Prove that the metric space (Z,) is complete.

A: Complete Metric Spaces: A metric space (X,d) is said to be complete iff every Cauchy sequence in X…

Q: For a normed space X, prove that the dual of X is separable implies that X is separable. Is the…

A: NOTE: According to bartleby we have to answer only first question please upload the question…

Q: If x is a normed linear space and the closed Unit ball M is compact, then X is finite dimensional.

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Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is 1.

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Q: 1. Let X C R"-1 be bounded. Prove that X considered as a subset of R" is a volume zero set.

A: Given, X⊆ℝn-1 That is, X is a subset of ℝ in terms of exponential n. So, let we consider the set,…

Q: Prove that any smooth manifold (M, A) has a countable smooth atlas V (so that D(V) = A)

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.