Let R be the set of real numbers and d: R x R → R be defined as follows. 5|x-y| d(x, y) = 3+2|x-yl Then (R, d) is a metric space. True O False

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: Ex: Let (X, d) be a metric space and E, andE2 EX Show that whether each one of the following…

A: According to guidelines I solve first 3 subparts so kindly request you to repost the remaining…

Q: 4:: what the Null Space of is T;: R3 R³( T,x= xx a) R, R (T = X.Q

A: As per guidelines, we will solve first three sub-question: We have to find the null space of  1.…

Q: (b) Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or…

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Q: Q. Assume that (R, T) is the usual Euclidean topological space. D= {(x, y) E R?, y = x² +3} C R.Then…

A: Solution: Here the subset of ℝ2 is D=x,y∈ℝ2,  y=x2+3 Now, The set D contains points x,y which…

Q: Let X and U be Banach spaces and let C: X → U be a compact linear map. Assume that U is infinite…

A: Given that X and Y be Banach spaces and let C:X→U be a compact linear map. and let U is infinite…

Q: (1.) Prove that (R", d) with d : R" × R" → R defined by d(r, y) = E(Ik – yk)² k=1 is a metric space.…

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

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: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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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) = 5|x10 - y1º|.…

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

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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|x - y®I. Then (R,…

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

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: Prove the following: Suppose that {X; : i e I} and {Y; : j € J} are indexed families of topological…

A: let us define ϕ:∏i∈IXi→∏j∈JYj=∏i∈IXα(i) by ϕx1,x2,x3....=xα(1),xα2,..... For any x=x1,x2,.... and…

Q: Prove the following: Suppose that {X; : i € I} and {Y; : i E I} are indexed families of topological…

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Q: Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a…

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Q: Q. Assume that (X,T) is a topological space. If A = {m, n, p} CX is a closed set. Then one of the…

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Q: 5. Let (X,r) and (Y,A) be two topological spaces. Prove that a function f:(X.r)(Y.A) is closed if…

A: Since you have posted multiple question according to company rule we are supposed to give solution…

Q: Suppose (X, d) is a metric space and let r> 0. Show that p: X2 → [0, c0) defined by p(x, y) = r·…

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Q: Set (Q, M, µ) a space of o-finite measure. we define the aplication 4: L° (Q) → L'(Q)* given by =…

A: Let g,h∈L∞(Ω and α∈F Claim:- ϕ(αg+h)=αφ(g)+φ(h)

Q: Q8. Assume that (R²,T) is the usual Euclidean topological space. D = {(x, y) E R², y = x} C R².…

A: Given below the answer in detailed

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

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Q: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

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Q: (f) The set B = {(x, y) E R² : x² + y² < 1} as a subset of R? with indiscrete metric is…

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Q: Q2: In Euclidean metric space (R², 1. 1), is the set A = {(x, y): xy < 1} compact? giving the…

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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: A. Let H be the set of all points (x, y) in R? such that x2 + 3y2 = 12. Show that H is a closed…

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Q: Let X =  R x R  x R = R3 and define the metric d: R 3 x R 3 to R  by d((x1, y1, z1), (x2, y2, z2)) =…

A: Consider X=R×R×R=R3 Define the metric d:R3×R3 to R  by d((x1,y1,z1),(x2,y2,z2))=x1-x2+y1-y2+z1-z2…

Q: Q. Assume that (R,T) is the usual Euclidean topological space. Let E = {0, ±2,+4, +6, ..} CR be the…

A: If dE is the notation for derived set of E and deltaE is the notation for boundary of E, then option…

Q: A. Let H be the set of all points (x, y) in R2 such that x2 + 3y2 = 12. Show that H is a closed…

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Q: Let R be the set of real numbers and d: R x R R be defined as follows. -> 2|x-y d(x, y) = 3+|x-yl…

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Q: 3. Find the closure and interior of the set A = {(x, y): xy = 1} as a subset of R? (equipped with…

A: Closure and interior

Q: Let (X, d) be a metric space and B = S(xo,r) be the open ball with centre at xo E X and radius r> 0.…

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Q: 4. Let X and Y be metric spaces and f : X → Y be continuous. Let SC X. Show: (b) If S is compact,…

A: First, we prove that fS¯⊆f(S)¯. Given that f is continuous and S¯ is compact, we can say f-1Y-f(S)¯…

Q: 8- Let X, Y are metric spaces and f: X → Y .If for any A C X, f(A) C f(A) then f is continuous.

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Q: Let 1={Ø}U{UER:NC U}, -n+3 ). A = :n EN and B = [0, ∞). Then n+1 in the topological space (X, 1) O…

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Q: Let (S, d) be a metric space and suppose that ρ:S×S→R is defined by ρ(x, y) =d(x, y)/(1 +d(x, y))…

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Q: d(r, y) Let (X, d) be a metric space and let p(r, y) = for all r, y E X. Show that (X, d) 1+d(x, y)…

A: According to our company policy we are supposed to answer only first question i.e. (4) 4) in this…

Q: Let A be a non-empty subset of the metric space (X,d). show that a. d(x, A) – d(y, A)| < d(x, y)For…

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

A: ANSWER :

Q: Let {Xa | a E A} be a collection of Hausdorff topological spaces. Prove that IαEA Xa is Hausdorff…

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Q: Let X and Y be metric spaces. Assume that E is a dense subset of X, and f, g: X Y are continuons…

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Q: Let V be the set of all real numbers. If x and y are elements of V and a E R, define: хеу-ху+5 a O x…

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Q: O let d:Rx R R be defned byd(x)= (x-, Ha,ysR (Rd) is a wmetric space. Rd) Ihen

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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: Let V be an inner product space. Then for x, y, z ∈V and c ∈F, the following statements are true.…

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Q: Let f: X, Y be a continuous surjection between metric spaces. If X is compact then A. X is complete…

A: Since X is compact , so it is complete. A option is true.

Q: 2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined by ρ(x, y) = d(x, y) 1 +…

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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.