if (X,d) is discrete metric space, ACX and A+D (x,d) Find the sets of A° and A (functional analysis )

Q: Show that the function u(z, t) : satisfies ua = %3D %3D a2t2-2

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Q: (a) Define the T1-space.

A: Define the T1 - space

Q: 3.4 Give the set of limit points Ao of a singleton A {(5, 2)} on the plane R2 with the discrete %3D…

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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: Let A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

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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: 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: c) For x e ll n2, define || x ]| = || x || 1+ || x ||2: Show that (11 n2, ||· || is a Banach space.

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Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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Q: Let X = (1, 2, 3), Tx = (0, X, (1), (2, 3}}, then (X, Tx) is %3D %3D 1) a T1 space 2) a T2 space

A: T1 space: A topological space X has the T1 property if x and y are distinct points of X, there…

Q: Example (4): Let (M, d) be a metric space. Define a function e : M X M → R as: e(x,y) d(x,y)…

A: Metric space is the base of topology. A metric space is a function that maps a non-empty set under…

Q: 10. Let X be an inner product space and T: X→→→→→X an isometric linear operator. If dim X<∞, show…

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Q: Prove that linear maps are bounded for multivariable calculus

A: In this question, concept of linear map is applied. Linear Map The matrix of the linear map can…

Q: Y Show that f (2)= 2+ Con formal maps %3D on %3D

A: Given that,f(z)=z+1zz=1

Q: consider the space generated with the set of functions  {x, x^2} , show that the set of elements of…

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Q: Give the set of limit points A0 of a singleton A = {(5, 2)} on the plane R 2 with the discrete…

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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: The Hamacher product is the t-norm given by ab t(a,b) = a +b – ab Show that this function satisfies…

A: Here the Hamacher product is the t-norm given by t(a,b)=aba+b-ab We have to show that this function…

Q: Q/ if (Xi, Txi) and (Yi, Tyi) are topological spaces then if X₁ Yi Prove that X₁ xX₂Y₁xY₂ Where…

A: We will use the basic knowledge of topology and set theory to answer this question correctly and…

Q: Prove Theorem (AT )T = A

A: Given: Theorem, ATT=A.

Q: Show that if a non-zero element in a normed space X, then there is a bounded linear functional f on…

A: We need to show that if a non-zero element in a normed space X, then there is a bounded linear…

Q: Let V(F) be an inner product space and a, Be V. Show that a= B iff ( α, γ)- (β,)V γε V.

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Q: Let X and Y be compact spaces. Then X × Y is compact.

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Q: Show that if a Banach space X is non-trivial, X + {0}, 3). then X' # {0}.

A: According to the given information, it is required to show that:

Q: Let V be a real inner product space, and let u, v, w E V. If (u, v) = 1 and (v, w) = 3, what is (3u…

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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: 0:1 Show that norm ||x|| = (Elx¡l?)ž satisfies the conditions of normed space. %3D

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

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Q: Exercise.3 Show that the following the functions d and p on X = R? are metrics or not, such that 2.…

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Q: Theorem 7.2. LetX,Y be topological spaces and yo E Y. The constant map f : X → Y defined by f(x) =…

A: Given, X, Y be topological spaces and y0∈Y. The constant map f:X→Y defined by fx=y0 Shows in the…

Q: If f(x) and g(x) have a period p, show that h(x)3Da f(x) + b g(x) (a, b constants) has the period p.…

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Q: c spaces. Show that X x Y and Y

A: The sup-metric d1 on X×Y is defined as d1x1, y1, x2, y2=max dx1, x2, ey1, y2. Similarly, the…

Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

Q: Let F(x, y, 2) = (z, 1, y). a. Find divF and curlF.

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Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

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Q: (a) Show that every metric space is T2-space.

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Q: Use Green's Theorem to evaluate F dr, where F(2, y) = (9xy, y" - 8) and C is the rectangle with…

A: That's easy. Have a great day!!!

Q: (a) Let X and Y be normed spaces and F :X→Y be a continuous linear map. Show that F is closed . Is…

A: Given that,                   X and Y are normed spaces and F : X → Y is a continuous linear map. We…

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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Q: Consider the space L'[0,1]. Let p(f) :

A: Given, pf=∫1234f212, f∈L20,1. Standard L2-norm is f2=∫01f212

Q: Find a linear functional ℓ on R3 that satisfies the following conditions: ℓ(1, 2, 2) = 1 ℓ(1, 2, 3)…

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Q: 8- Suppose that X,Y be a normed spaces then the function f: X → Y is continuous at a point xo E X if…

A: True

Q: 4. Show that the function f : X → Y defined by f(X) = X U {a} vX €x is a bijection.

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Q: Show that if X and Y Hausdorff spaces, then so is the product space X × Y. are

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Q: Prove that dual space of R^n is R^n( its Functional analysis topic)

A: to prove dual space of ℝn is ℝn. proof let the dual space of ℝn be denoted as ℝn*=M we will show…

Q: We have seen that the space C-1, 1] with (f, is an inner product space. Let U = {f € C [=1,1] :…

A: Here we will show the required result.

Q: Q#1: Discuss Application of inner product spaces with examples.

A: Important Application of Inner product spaces is to find Least squares approximations for a…