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Q: ) Consider the sequen ces equi' pped space f absolutely ce uith He Ofalloing' onvergent norm (0.1)…

A: We will use the basic knowledge of norms and functional analysis to answer all three parts of this…

Q: 1. Prove that for every non-gero normed space X there is a non-zero bounded linear fmetional acting…

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Q: H is a Hilbert space, A: H → H is linear ar = for all r, y e H. Use th closed graph theorem to show…

A: For two normed spaces V and W, a linear operator T:V→W is closed if for each xn∈V, xn→x and Txn→y…

Q: Q3 Assume that (M,d) is a Compact metrie Space. Show that : (M,d) → (ya) is Continous bjective then…

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Q: A point y in space either belongs to a given closed c ough y so that whole of the x lies in one open…

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Q: f(x,y) = (In(y + x²)) x² + y² D = {(x,y) ERIy2x²-1 and y ≤4-e-*) (b) Observe the set Is D…

A: (b) The given set is D=x,y∈ℝ : y≥x2-1 and y≤4-e-x To find: The given set is bounded or closed or…

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: Provethat amappi'ng from a metric SpaceXto metri'c space X is con tinuok ifand only f the inverse…

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Q: Let X = Rand def ine |-|| on X as ||r||= V then X is not normed linear space O True False

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Q: Let (X.d) be mnrtric space and X is closed then X is O complete Ono decsion Onot complete

A: A metric space X is said to be complete if every cauchy sequence in X is convergent. That is, every…

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: 5. Let V be an inner product space, and T E L(V). Prove that null(T) = (Im(T*)). %3D

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Q: Show that every plane through the origin in R³ may be identified with the null space of an element…

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Q: The space X = {0,1} with topology T = {0, {1}, {0, 1}}) is To but not T Select one: O True O False

A: Discrete spaces are always T0 , So X is T0 .  A space X is a T1 space if and only if  for each x∈X,…

Q: Date. AC) Lat normed linear space X, Y= spomE and aex only if fla) = 0 wheneres f EX' omd f30…

A: To show the necessary part- To show that for each f∈X' whenever f vanishes on E, i.e.if f=0 then…

Q: Let X = R and def ine ||.|| on X as ||x|| = x² then X is normed linear space True False

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

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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: 5. Let (X, d) be a metric space. i) Show that the open ball B, (ro) = {x € X : d (T, xo) < r} is an…

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

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: 2/Define the topology If X-R² and T = {GK, KERZULAR} GK = {(XY) ER²|x+y >K3 Prove is topolgy on R²

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Q: let (x, m) be a measurable space. then ¥8:% [, o] are measural ble funetion, then ) I +9 is mea…

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Q: rove that spaces Ip for os p<l If for pe(0, 0) {0} and complet. are

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Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If ACB,…

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Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

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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: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

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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: 8

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Q: showthat two metricspace s (Xyd) and (X,d) are home omorphicifand onlyifevery open…

A: This is a problem of Metric space and Topology.

Q: Assume that (M, d) is a compact metric space. Show that if f: (M,d) → (Y, d) is continuous and…

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Q: 8. let C be the curue space ret) = (e*,2et ,2et ),0s ts lnca) the rength of C is

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Q: E be a subset of a normed space X and Y = Span E and a € X. Show that Let a € Ỹ if and only if f(a)…

A: To find: a∈Y¯, and f(a) = 0 whenever f∈Xι and f is 0 everywhere on E.

Q: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

Q: IfS is a subset of an innerproduct space V(F) then show that Scs=(0}. %3D

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Q: Let S = {u, v, w} be an orthonormal subset of an inner product space V. What is ||u + 2v + 3w||2?…

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Q: Q2: In Euclidean metric space (R2, 1. 1), is the set A = {(x, y): 2x² + y² = 1} compact? giving the…

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Q: Every T2 (Hausdorf f) space is T1 space. True O False O

A: Since T2 is a product preserving topological property. So T2 space is a T1 space.

Q: If X is an inner product space and A = {0}, then A = %3D %3D O X O A

A: A⊥ = X

Q: -Re call, Ac (Xicd) AB .bounded if JxcX,ryo Sucb that Ac BCX) A B(メイ) property (Xdl) methic space…

A: We will use the definition of compact set and bounded set to prove this property.

Q: Q3) Let L = Z* and |x ||= max {]x1 - x3l, x2l, 15x,1} Vx = (x1, X2, X3, X4) E Z*. Is (Z*, ) is a…

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Q: 7. Let (S, p) be a metric space and T : S → S be a contraction mapping. Prove that T is uniformly…

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Q: tet is cosinte toploy on set R show that (R, t) is O NOVmal space- Regualra space

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Q: Q6: (a) In any metric space (S, d), prove that every compact subset T is complete. (b) Prove that…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

Q: ii). what { disie [4NI} to fobrm vedtor space ? set vectas a bauis for en N-dimeneional alte the…

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Q: Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only…

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Q: 3 \et F be anormid space and- |2+j|l <llxll+yllprove that F has a stridtl (3) y convex- orm

A: We know that a normed space is strictly convex if and only if x≠y, ∥x∥=∥y∥=1, we have ∥x+y∥&lt;2 Let…

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

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Q: Use th properties of linear Space A upon t to prove s a) 0.X - o for all X EX b) a.x=0 =>(a=07v…

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