) Assume (X,A) is a Ealgebra and ld ycx be an anbitrary subset. Show that the Pami ly of sets 3{unyluent é-algebra on y B:= iう a

Q: show that the discrete metric on u Vector space X+3 of eannot be obtained from a norm.

A: To Show: Show that the discrete metric on a vector space X ≠ 0 cannot be obtained from a norm.

Q: 3) Let V = {("), a,b ER'} (note R* = {r € R|r > 0} ). Define "addition" on V by C) + C) - G)…

A: Given: Let V=ab, a,b∈ℝ+ ; ℝ+=r∈ℝ / r>0 Define vector addition as, a1b1+a2b2=a1a2b1b2 Define…

Q: Find a base for the Standord tapology-

A: This is a question of topology.

Q: s paces Xand y, then both T and Ta boanded op erators are

A:

Q: 4) Define on the cartesian product of sets R:=R x R = {(^, z) | A, z E R } an addition and a…

A: We first show that R is a commutative ring. That is we show that R is commutative group under…

Q: 7. Let M be the set of all n x n matrices. Define a relation on ~ on M by A B there exists an…

A: A relation ~ is said to be an equivalent relation if it hold following three properties (1)…

Q: Question Show closed bale Y in Mebic space (X,d), Alsa ఎవక is set that ? then shaw is. complete d)is…

A: Given: Y is closed ball in metric space (X, d). To show:  1. Y is closed set in (X, d). 2. If (X, d)…

Q: 2.x²y"+xy'–3y= 0,x>0

A:

Q: 25. For any set S CR, let S denote the union of all the open sets contained in S. (a) Prove that S…

A: The given question is solved below.

Q: 1) Every convex set of a vector space X is affine set.

A:

Q: Let Xis Hilber @ 1| + |1

A:

Q: let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B…

A: Let (X,t)  be  not  connected.We  have  to  show  that  X=A∪B ,where  A and B are  elements  in t…

Q: By example prove that a Housdorff space with a countable basis need not be metrizable.

A: We know that a topological space is metrizable then it is regular. For example, Let ℝ be a set of…

Q: Show that for any two finite-dimensional L-representations V and W of L: (a) Hom (V, k) ~ V* as…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

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

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: Show that finite dimenshional inner product space is hilbert space

A: An n-dimentional inner product space is not Just Topologically isomorphic to Rn. It is isometric to…

Q: If A is a bounded linear operator on a H, if Hilbert space H, prove that R(A*) = and only if A is…

A:

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

A:

Q: Show that the set A = {fm,b : R → R | m # 0 and fm,b(æ) group under composition of functions. (The…

A: To Show- Show that the set A = fm,b : R → R | m ≠ 0 and fm,bx = mx + b, m, b ∈ R forms a group under…

Q: 5) Let A= (2,3) U {4} be a subset of R with the standard opology. Find Ə(A).

A:

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: 5. Let V P2. For any p = ao + a1x + a2x² and q = bo + b1x + b2x² in V, define (p, g) = aobo + 2a1bı…

A:

Q: Let A€BL(H) be a compact operator, then show that A*A is a compact

A:

Q: 3. Consider the set e = {x = (r1, x2, x3, ...) E C* : i=1 consisting of all absolutely summable…

A:

Q: Use membership table to prove (AunionB) – C = (A – C) union (B - C).

A: We need to prove the following statement using membership table. A∪B-C=A-C∪B-C

Q: Define complemented lattice. Let (L, *,Ð, ', 0,1) be a Boolean algebra. Then show the following:…

A: Given: (L,*,⊕,',0,1) is a Boolean algebra.We need to define complemented lattice and show that…

Q: The region bounded by f(x) 2x² + 12x + 14, x = = 0 is rotated about the y-axis. Find the volume of…

A:

Q: Suppose that N = (-∞, ∞), and M = {(-∞, 1), (0, ∞)}. It is easy to check that the minimal o-algebra…

A: The objective of the question is to understand the origin of each member of the sigma-algebra Fm,…

Q: direct and direct let be a k. then pooduct of is z'G) product ZCH) and zck).

A: To show that Z(G) is a direct product of Z(H) and Z(K) when given that G is a direct product of H…

Q: bijective Show That There is map from CCA) to PCA).

A: Sol:- Let A be a set and let C(A) denote the set of all functions from A to {0,1}, which is the…

Q: Show that is not compact for the Standard topology but is compact for the cofinite topology.

A:

Q: a Normed &pace and and CL(X,Y) denste, of bounded Let X,Y be a Suppose BL(X,Y) respectively, the…

A:

Q: 5. Show that the function that sends an ordered pair ((x1, x2, x3), (Y1, Y2, Y3)) of elements in R³…

A: I have write answers below Explanation:

Q: Determine whether each product is an inner product space in R3. If not, explain why. 1) = u1v1 +…

A:

Q: Calculate all conjugacy classes for the quaternions.

A: We know that group of quaternions is given by, Q8=1, i, j, k, -1, -i, -j, -k. We know that two…

Q: Prove disprove that (R.Tco-finite) is Te-space or

A: To Prove - Prove or Disprove that ℝ, τco-finite  is T2 - space.       Definition used - T2 - space…