Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric                      d((x1, x2), (y1, y2)) = max{d1(x1, y1), d2(x2, y2)} is also separable space.

Q: Draw the Taxi-metric hyperboloid |x| + |y| + |z| = 1

A: As per the question we are given the equation of Taxi-metric hyperboloid as : |x| + |y| + |z| = 1…

Q: (a) Define the T1-space.

A: Define the T1 - space

Q: 4. Check that the following are vector spaces: (a) RN. 55 / 600 (b) CN. (c) C(N), where 2 is a…

A: A non-empty set V, +, · is said to be a vector space over a field F together with operations of…

Q: Consider the inner product space (P2, (, )) with (p, q) = p(to)q(to) + p(t1)q(t1) + p(t2)q(t,),…

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Q: Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norn of function f(x)…

A: This is a problem of functional analysis.

Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

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: Es A (a), B (b), C(c) and D (d) are related as x a +yb + zc + wd +z+w=0, where x, y, z and w are…

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Q: Is the set V = {(x, y) x >0} where (x1, y₁) + (x2, Y₂) = (x₁x2, y₁ + y₂) and c (x, y) = (x,cy) a…

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Q: (b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ?…

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Q: Let R be the set of real numbers and d: R x R → R be defined as follows. d(x, y) = 3|x8 – y®|. %3D…

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Q: 1. Is V = why not? 1:0 20 and b≥( a vector space? Draw the graph and explain why or

A: By the answering guidelines we have to do first question is multiple questions are given so I am…

Q: Thm. Let v be an V be an inner product Space V and then жує | 13 11211. 11111 further equility is…

A:

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

A:

Q: (3) Let R² and R be endowed with their Euclidean metrics. (a) Show that projections p₁(x, y) = x and…

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Q: - theorm; – Every inner product space is a norm linear space

A: We have to prove that every inner product space is a norm linear space.

Q: Let x = (13, −20, 15), v1 = (1, −2, 3), and v2 = (−1, 1, 1). Then the set {v1, v2} is orthogonal. It…

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Q: Every T3 space is regular space. True False

A: Since a regular T1 space is called a T3 space.

Q: Let U = ((1, 1, 0, 1), (0, 1, 2, 3), (2, 3, 2,−1)) and W = ((1, 3, 4, 4), (1, 1, 0, 2)) be sub-…

A: The given U=1, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, -1 and  W=1, 3, 4, 4, 1, 1, 0, 2 be a subspaces of ℝ4.…

Q: Which of the following are vector spaces? Justify your answer. (a) The set of all polynomials of…

A: Here we check the part (d) and (e) is vector space or not.

Q: Q4. Consider the topological space (X,T), where X = {a, b, c, d) and T = {o, X, {a}, {b}, {c},…

A: Given: Q4. a)  X, T is a topological space, where X=a, b, c, d and T=∅, X, a, b, c, {a, b}, {a, c},…

Q: ->> Points A (a), B (b), C(c) and D (d) are related as xa + y + zc + wd = 0 and x+y+z+w=0, where x,…

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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: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: Show that V={(x,2020x): x in R} is a vector space

A: According to the given information, it is required to show that the given set is a vector space.

Q: real line ℝ and complex plane ℂ  are complete metric space

A: Definition  Complete Metric Space : A metric space (X,d) is called complete if every Cauchy Sequence…

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

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Q: Thm. Let V be an inner product Space and ждек then | 1 ≤ 11all. 1171/ further equility is hold iff…

A: Using the properties of inner product And  aā=|a|2 , for any complex scalar  ||cx||=|c| ||x||, for…

Q: r', where i =

A: As per rule, allowed to answer first question and post the remaining in the next submisison.

Q: Theorem 9.13. Let (X, d) and (Y,e) be metric spaces. Then X × Y is a metric space.

A:

Q: Let R" be a real vector space, and u = (u1, u2, Un), v = (v1, V2, ..., Vn) E .... R", then show that…

A: For a real vector space, the inner product . , . satisfies the following properties a)  u+w , v =…

Q: Let E and F be two vector spaces with the same finite dimension dim(E) = dim(F) For every linear map…

A: Follow the steps.

Q: Q1: Let M = R² be the set of all ordered pairs of real numbers and d: MXM →R be a function defined…

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: (b) Let X bę a normed linear space. Let Вх, г) %3D (у є Х: || x-у || <r). Show that B(x, r) = {y e…

A: To show - The ball B¯(x, r) is closed. Indeed, for any given y ∈ M \ B¯(x, r) we set R = d(x, y)…

Q: Q7: Prove that ( IR", II-||) is complete metric space.

A:

Q: (c) If X={1, 2, 3}. T= {4, X, {1}, {1,2}, {3}, {1,3}}. JUSTIFY whether the topological space (X, t)…

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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.