Q3 Prove : If V is a normed linear space, and WcV, and U a closed subset of W Then given ve V,3u" eV, such that v-u <|v-u|| for all ueU.

Q: Q2) Let (A,T)be topological spaces; i=1,2,3. Then A, x A2 x Az is homomorphic to (A, x A2) x A3 I

A: Definition : Let X and Y be spaces, and let B be the collection of sets U × V , where U ⊆ X and V ⊆…

Q: 1. Consider the set V of all polynomials of degree two or less, along with the following definitions…

A:

Q: (Proving Linear Algebra)

A:

Q: 1. Provide an example of a vector space Mmxn(Q) that is isomorphic to the vector space P19(Q).

A:

Q: Let V is finite vector space and suppose that V=internal direct sum of U and S1. Also V=internal…

A: Either S1 and S2 will be disjoint or identically same. If S1 is a subset of S2 then S1= S2.

Q: 5. Let V be an inner product space over C. Show that for any v, w e V (v, (v, w)w) = | (v, w) |² and…

A:

Q: 5. Let V be a finite-dimensional inner product space over C. Given u, v E V, prove that ||u+ v||? –…

A:

Q: 6. Consider a normed vector space S = (R,|- |). Show that, for all r, y e S, |z - y| 2 | – l9|-

A:

Q: 5. Let V be an inner product space, and T E L(V). Prove that null(T) = (Im(T*)). %3D

A:

Q: 2. Let M be a collection of subsets of a metric space X. We say that M has proper AUBE M whenever A,…

A: 2. Given, T is the property of A,B∈M such that A∪B∈M and A∩B≠∅. i.e. T is the property such that for…

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: Q6: Show that R³ = {(x1,x2, x3)|lx1,x2, x3 E R} is a vector space

A: Vector space is the field created by set of vectors with specified direction and magnitude, usually…

Q: Suppose V is an inner-product space and u, v e V satisfy ||u|| = ||v|| = 1 and (u, v) = 1. Prove…

A:

Q: Q7. Assume that (R,T) is the usual Euclidean topological space. N = {1,2, 3, ....} C R. Determine…

A: There is no interior point of N.

Q: 5) Let V, W be finite dimensional vector spaces over F. (a) ( Prove that (b) Use this to prove that…

A: Question is wrong • please see the justification

Q: Let X = Rand def ine || || on X as |a|= min{|a,0) %3D then show that if X is a normed linear space…

A:

Q: Let X and Y be normed spaces. Then show that the following hold : (i) If Y is a Banach space, then…

A: We consider an arbitrary Cauchy sequence (Tn)n≥1 in BL(X, Y) and show that (Tn)n≥1 converges to an…

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

A:

Q: Consider the Hilbert space H = Let (X, , μ) be a o-finite measure space. L²(X, N, μ) =: L² (µ). If ¢…

A:

Q: (2) Let V be an inner product space. Let u and v be nonzero vectors in V such that for every w E V,…

A:

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: Let X = Rand def ine ||.|| on X as ||x|| = 1 %3D then X is not normed linear space True O False O

A:

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: 1. Show that V = R² is not a vector space over R with respect to the operations: (a, b) + (c, d) =…

A: A real vector space is a set V with the special element zero, and three operations: (1) Addition:…

Q: (a) Let || • ||, and || • ||, be norms on a linear space X such that for some r > 0 || x |l, <r || x…

A:

Q: 10) Every normed space is (a) Metric Space (b) Vector Space (e) a and b (d) No Answer. a b O c O d O…

A: This is a multiple question, i.e., two separate and unrelated questions. So, as per company policy,…

Q: Q3 Assume that (M,d) is a Compact metric Space. Show that, f: (M,d) → (Y, 2) is Continous bjective…

A: f:(X, d) to (Y, d) is said to be homeomorphism if  1) f is continuous.  2) f is bijective.  3) f…

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

A:

Q: 8. let C be the curue space ret) = (e*,2et ,2et ),0s ts lnca) the rength of C is

A:

Q: Let X be a normed space and A € BL(X) be of finite rank. Then σe (A) = σ₁(A) = o(A).

A: 1

Q: 3.6.3. Let ||· ||a and || || be equivalent norms on a vector space X, and let E be a subset of X.…

A: Given: ·a and ·b are equivalent norms on a vector space X, and let E be a subset of X. To prove:…

Q: 12. Suppose that a, b e R and a < b. Show that the vector space C[a, b] of all continuous complex…

A:

Q: Q1) Prove or Disprove: Let L(F) be a linear space and A,B are two convex subsets of L. Then, AXB is…

A: The problem is related with convex set. Given that,                 LF  is a linear space  and A, B…

Q: 3.6.3. Let || ||a and || || be equivalent norms on a vector space X, and let E be a subset of X. For…

A: Given : .a and .b are the equivalent norms on a vector space X and E be the subset of X. To prove :…

Q: (a) What is the subspace topology R on Y induced by T? (b) List all of the R-closed subsets of Y.…

A: Subspace topology B on Y induced by I :                   B=(Y,IH)

Q: 3. Prove that in any normed linear space, xo is the Chebyshev center of the closed ball {x : ||x –…

A: Given: Given any normed linear space x

Q: (13) Show that | l is a norm on the space L [a,b].

A: Please check step 2 for the solution.

Q: Let S = {u, v, w} be an orthonormal subset of an inner product space V. What is ||u + 2v + 3w||2?…

A:

Q: (b) Let X be a normed space and Y be a Banach space. Define the operator norm on the linear space…

A: (b) It is given that X is a normed space and Y is a Banach space. Firstly, define the BX,Y  as,…

Q: Prove the following statement or give a counter example showing it is false: Let V be a vector space…

A: We have to solve given problem:

Q: Let V and W be finite-dimensional vector spaces and T: V →W be linear. (a) Prove that if dim(V)…

A: (a)Let B={v1,…,vn} be a basis for V. Then, C={T(v1),….,T(vn)} spans T(v)

Q: Let X=R and T = {]a, +oo[/a R}U{o,R}, then (X, 7) is a regular space. True False

A: This is a problem of topology.

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: Let E be a Euclidean space with inner ratic form. For all u, v E E, we have the C y(u, v)² ≤…

A:

Q: (b) space and X+{ 0 }. Show that B< (X, Y) is a Banach space. Let X be a normed space and Y be a…

A: X be a normed space and Y be a Banach space We have to show that B(X, Y) - set of all boundes linear…

Q: A Consider E = \R as a vector space. Show that: if and only if fixed real winber A function f: E →…

A:

Q: Let v be a vector spacel and Ź Vi,i. Vn& a Maximal set af linearly independent elements Of . Then…

A: let ,    Let S= {v1,v2,....,vn} We are given thet S is a maximal linearly independent set vectors of…

Q: E 6. If W, and W2 be any two sub-spaces of a vector space V, then inWa is also a sub-space of V in…

A: