3. Prove that in any normed linear space, xo is the Chebyshev center of the closed ball {x : ||x – xo|| < r} Prove also the uniqueness of the Chebyshev center.

Q: (4) Suppose V and W are finite dimensional vector spaces such that dim(V) > dim(W). Show that any…

A:

Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

A:

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

A:

Q: _I. Let H and K be Hilbert spaces and A E B(H, K). Show that A is compact if = and only if A* A is…

A:

Q: Let S = {A2«2| trace(A) = 0}, i.e. S = Show that -a S is a vector space.

A: The solution is given as

Q: Is it possible to make Z into a vector space if we redefine addition?

A: To check: the possibility to make ℤ into a vector space.

Q: Let V be an inner product space, and suppose that x and y are orthogonal elements of V. Prove that…

A: Given:

Q: Let (E,p) be a pre-Hilbert space with ass E, we have the Cauchy-Schwarz inequality |y(u, v)| ≤…

A:

Q: Consider the inner product space (P2, (, )) with the inner product (p(t), q(t)) =…

A:

Q: 5. Show that if S = {v1,V2, .. .} is a basic for the space vector V and c is a non-zero ... ... ...…

A:

Q: 1. The set V = {(x,y,z) ER:y = 0} with traditional addition and scalar multiplication is a vector…

A: A detailed solution is given below.

Q: Let V, W be finite-dimensional inner product spaces and T € L(V, W). Prove: (a) If T is injective,…

A: Injective Function:  A function is said to be injective if each element of the codomain is mapped to…

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

A:

Q: Consider the inner product space (P2, (, )) with the inner product (p(t), q(t)) =…

A:

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

A:

Q: Note: Unless otherwise stated, E stands for a locally convex space (LCS). 2.4.6. Suppose p and q are…

A: Given that E is a locally  convex space also given that p and q are seminorms on a  Locally convex…

Q: Let (M, d) be a metric space. Show that the set {B(x, 1/n); x E M; n e N*} is a Basis for the metric…

A:

Q: Theorem 16. Show that if t: V→ V is a self duct space V, then s* is is self-adjoint for every lin…

A:

Q: Consider the linear operator L: BC[0, ] → BC[0, 7] defined by Си — и" - 2и is unbounded in the sup…

A: Consider the linear operator L:BC0,π→BC0,π defined by: Lu=u''-2u. We know that the set BC0,π is…

Q: Let T be an invertible linearlo-perator on a finite-dimensionalvector spaceProve that T is…

A:

Q: If scalar multiplication and addition is redefined on R² in the following way, show that it is NOT a…

A:

Q: Consider the inner product space (P2, (, )) with the inner product (p(t), q(t)) =…

A:

Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

A: Given : V is any normed vector space and x is any non zero vector and α&gt;0. To find :  A real…

Q: 2. Let (,): P3 × P3 → R defined by (p, q) = p(0)q(0) + 2p'(0)q'(0) + 3p"(0)q"(0) + p''(0)q"'(0) for…

A: Given, ,:ℙ3×ℙ3→ℝ defined by, p,q=p(0)q(0)+2p'(0)q'(0)+3p''(0)q''(0)+p'''(0)q'''(0)  ∀p,q∈ℙ3 a) To…

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: Theorem 4.7. (1) A T2-space (Hausdorff) is a T1-space. (2) A T3-space (regular and T¡) is a…

A: 1)  Consider X is a T2 - space. To prove that, X is a T1- space. By the definition of T2- space,…

Q: Let V be an inner product space, and suppose that x and y are orthogonal vectors in V. Prove that…

A: Let V be an inner product space.Let x and y are orthogonal are orthogonal vectors in V.

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

A:

Q: Suppose V is a 4-dimensional vector space, and that T is a linear operator on V with minimal…

A: Dimension of Ker T

Q: Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the…

A: Let X, d  be any metric space. It is required to prove that there exists a metric d such that d and…

Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

A:

Q: Let V be a K-vector space and U1, U2, U3 three sub-vector spaces of V. Show that: dim(U1) + dim(U2)…

A: first of all we must know about the dimension of the vector space.what is the dimension of the…

Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

A:

Q: Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1…

A:

Q: Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = &lt;x, y&gt;z for all x…

A:

Q: ) For the vector space C[0,1], define the inner product (f,g) = , f(x)g(x)dx. Let f = x + 1 and g =…

A:

Q: Theorem 7.32. Let X and Y be topological spaces. The projection maps tx,Ty on X×Y are continuous,…

A:

Q: (b) Prove that the unit ball of a normed linear space is compact if and only if the normed linear…

A:

Q: et H be a HIlbert space and S b of S is a closed linear space of

A:

Q: The vector spaces M2, 2 and M5,1 are isomorphic to each other ylgn ihi

A: False

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: Let X be a normed linear space such that its dual X' is separable, then show that X is separable ?…

A: This is a problem of Functional Analysis.

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: 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: 14. If d is a metric on a vector space X‡ {0} which is obtained from a norm, and å is defined by…

A:

Q: 7. Let V be a vector space and l: V → R be a linear map. If z E V is not in the nullspace of l, show…

A: Given: 7) V is a vector space and l:V→ℝ is a linear map. To show: If z∈V is not in the null space of…

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

A:

Q: Let X is a normed linear space and the ball M= {xe X:|| x ||≤1} is compact, then X is finite…

A:

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.