Let X + 0, |X|> 2. Then (X, Tna) is not To -space. Select one: True O False

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: Let r, y E R". Using the triangle inequality ||r + y|| < ||1|| + ||y||, show that r|| – ||y||| < ||r…

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Q: Let (R, d) be usual metric space, then Ris ot separable True False

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Q: Let (X, M, µ) be a measure space such that uX = 1, and let f : X → (0, 1). Show that u) / fdµ 2 1

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Q: Use the Squeeze Theorem to show that f(x) = x² is in R[0, 3]. %3D

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Q: Prove the following equation by showing the function is square-integrable on the real line, ||r±…

A: As the function is square-integrable we will construct a Hilbert space having inner-product…

Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: let H = {n² |n E z*}, determine whether it is closed under

A: Given : H={n2|n∈Z+}   To Check:  H is closed under binary addition or not

Q: Let x, y be real numbers such that xy=1x and x≠y. Prove that x^2+y^2≥2\sqrt{2}|x−y|.

A: Given: xy=1x,x≠y To find: x2+y2≥22x-y

Q: On what interval is In(x² + 1) concave up? くxく

A: This question is about concavity of a function

Q: atisfies the different aditions, f(0) = 0 %3D

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Q: Let (R. d) be discrete metric space, thend can not induces a norm on X O True O Fahe

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let X and Y be two T, spaces, then X×Y. :is

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Q: 4).ェ Pare subset, -f y such that CeD than fic) is not eD, then the Convers %3D そイue.

A: Solution As it's mention C=D and f^-1 same , so each and every element from  f inverse we get equal.…

Q: Show that R is not compact.

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Q: Let f and g represent differentiable functions such that f ″ ≠ 0 and g″ ≠ 0.

A: Concavity of a function can be determined using the second derivative of the function. If the second…

Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: Let R = [0, 1] × [0, 1] . Est each with at least six rectan IR 6(x + y) dA 2

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

Q: As the fun ction Leay) xy js Continuous I+ x*+y? e (-1, 2)

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Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

Q: Suppose n is an integer and z-z3D =r is a contour traversed n dz %3D once in counter clockwise, then…

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Q: Let ( Xr d be ame tric space,show that exist atopology on X induce b

A: Let (X, d) be a Metric Space We know that in a Metric Space  (1) arbitrary union of open sets is…

Q: Stafe it fre andwer, your er Let f and be cenfirious on a Space X into R."Then te poduct fxg…

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Q: Let x e R,-1. Prove that VNEN (1+x)" >1+ nx

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Q: Sketch the set of points satisfying: |z - 1 + 2i| = |z + 3 + i|

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Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

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Q: an open ball in the set X, with its radius & satisfying -d(x, a), Vx € X at the open ball Be (x)…

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Q: (a) Prove the following relation: В (x + 1,у): B(x, y) x +y %3D

A: We have give Beta function relation, we have to show left hand side is equal to right hand side.

Q: Show that f: R → R defined as f(a) = 3a³- 4 is a one to one %3D iuicuon.

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Q: Let (X, E, µ) be a measure space. Show that f is integrable if and only if fx ]f]dµ < o∞. Moreover…

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Q: Recall that for D a squarefree integer, the norm function N on Q[VD] is defined by N(a + bVD) = (a +…

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

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Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

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

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

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Q: Let f = u + iv be analytic in the connected open set and assume u and v satisfy u? + 3v2 = 4. Show f…

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Q: Let {x: 0<x <Tt} then x defined in interval [T,0) * True False

A: The given set is x : 0&lt;x&lt;π. Note that π is not included here.

Q: Let X = R, and M = {0}, then span (M) = %3D %3D OR O {0}

A: If, M = {0} then, span(M) = {0}

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

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Q: let Peas be delimed by Xく2 3-X I5 fixo continuous V= 2 2.

A: A function is continuous at a point if it's left hand limit is equal to right hand limit and both…

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: Consider the set R. Prove: (i) VxER, x 0 = 0;

A: To prove: For any x∈ℝ, x·0=0.

Q: Every T2 (Hausdor ff) space is T1 space. True O False O

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Q: Let X be a rs from Exp(0). Construct a (1 - a) CI for 0.

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Q: Show that the real number R with d(x,y)=|x-2y| are not a metric space

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Q: Qz\ show that ?_C=0} is not a swbopace of R

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Q: what sets X and Y is the function X Y defined by 3x f (x) = x +4 %3D ijection?

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Q: The solut (Туре yoL

A: This is a question on inequality from graph