Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an L-Lipschitz function. Then there is an L-Lipschitz extension. 3.

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, 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: Suppose f(z) is analytic in the unit disk D={|z| < 1} and |f(z)|<= 1 for |z| <1, Prove that…

A: Assume that f(z) is analytic on the unit disk D ={z: |z| < 1}.Also assume that |f(z)|≤1 for…

Q: where a, b e X, is also a metric on X. 2- Let (X, d) be a metric space and T the corresponding…

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Q: (a) Suppose that (M, d) is a metric space. Consider the mapping f: M [0, 1] be given and let di(x,…

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

A: Please check the answer in next step

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

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Q: Let (X, T) be a topological space and A,B CX- ConstYuct the general Yelation between Fr (IntlA) and…

A: For any subset B of a topological space X , Frontier or boundary of B is defined as  Fr(B) = CL(B) -…

Q: (a) Define a metric space (X, p). (b) Let p: R+ → R+ be a function defined by ln (2). Prove that p…

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Q: Suppose V is an inner-product space and u, v e V satisfy ||u|| = ||v|| = 1 and (u, v) = 1. Prove…

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Q: E be a Euclidean space of dimension (E(1, f)) = dim(Ker (f- id)), then s minimal.

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Q: Consider E =IR as a vector space. Show that: A function f: E → IR is a norm if and only if fu) =klxı…

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Q: (b) Let (M, d) be a metric space. Define a function e: MXM-R by e(x, y) = min {1,d(x, y)}; vx,y E M.…

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Q: Let X be a normed space, Xn, Yn EX such that Xnx, Yny, then 1-Xn + Yn → 4- ||xn-ynll→→llx - yll.…

A: Given that X is a normed space, xn, yn ∈X such that xn→x, yn→y. To find1. xn+ yn→x+y 2. λxn→λx ∀λ∈F…

Q: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

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Q: Let f be a continuous mapping of a metric space X into metric space Y,g be a continuous mapping of…

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Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th 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: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

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Q: Complete the proof of Theorem 6.2. Theorem 6.2. Let V be an inner product space over F. Then for all…

A: Let V be an inner product space over F. Then for all x ,y € V and c € F,For part (a)It is required…

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: * Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((r, y)) = x…

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Q: Let (X, d) be a compact metric space and let A ⊆ C(X; C) = {f : X → C; f is continuous} be an…

A: please see the next step for solution

Q: 3. i) Let (X, d) be a metric space. If æn → x in (X, d) and y e X, prove that lim d(xn, y) = d (x,…

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Q: 1: Let (X,) be a metric space and let d1(x.,y)=K d(x.y), K>0. Prove that d1 is a metric function on…

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Q: Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the…

A: Let X, d  be any metric space. It is required to prove that there exists a metric d such that d and…

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: For a metric space (X, d) and a continuous functionf from X into itself, show that the set of points…

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Q: Use the Completion Theorem below to show that the Completion of the discrete metric space X is…

A: By completion theorem, there exists a complete metric space which has a subspace that is isometric…

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, T) and (Y, T1) be topological spaces and f a mapping of X into Y. If (X, T) is a discrete…

A: For proving f is continuous enough to prove that inverse image of every open set is open.

Q: Let (X, d) be a metric space. Define dˆ : X × X → R, by: ˆ
d(x, y) = min{1, d(x, y)}. (a) Prove that…

A: To verify that d^ satisfies the axioms of a distance function

Q: 3 Let (Y, p) be a metric space. Give an example metric d on Y such that every finction f:(Y, d)→(Y,…

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Q: If the unit sphere {xe X : x = 1} in a normed space X is complete, prove that x is complete.

A: Given that X is a normed space. Let S1x=x∈X| x=1 be unit sphere of the normed space X. Given that…

Q: Let X,d) be a metrie space. Define d': Xx X >R by: d'Cxoy) Show that (X.d') is min (dl8), 1). a…

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Q: Let X be a compact subset of R and let C(X) denote the set of continuous real-valued functions on X…

A: let X be a compact subset of ℝ and let CX denote the set of continuous real valued functions on X.…

Q: 2.9.11 Prove that every function f: Z→ F is continuous, where the metric on Z is the restriction of…

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Q: Let (X, || · ||) be a Banach space, J CR be bounded and F C C1 (J, X ). If sup ||f(x)|| < , f€F, rɛJ…

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Q: Given the family C[0, 1] of continuous functions ƒ : [0, 1] → R, and the norm ||f|| = sup f(x)]. x=…

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Q: Let X, Y be normed linear spaces. On X x Y define || (x,y) |l, = || × |lx + || y lly and + || y ily…

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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: Let (X, d) be a compact metric space and T: XX be a mapping that satisfies d(Tr,Ty)<d(r,y) for all…

A: What is Metric Space: Metric space is an abstract set in mathematics, particularly topology, with a…

Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: if x is a compact Metric space and F e C (X₂Y) is equicontinuous, then F i's uniformly…

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Q: Consider E = C( [o,(]) as normed spaces with normS II f | = sup 1fcxl and l fll,=Sifwildx. %3D Show…

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Q: Let X is a normed linear space and the ball M= {xe X:|| x ||≤1} is compact, then X is finite…

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Q: Prove that any smooth manifold (M, A) has a countable smooth atlas V (so that D(V) = A)

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Q: Let f: (N,d) (N, d), where d is the usual metric and di is the discrete metric, defined f as f (x)…

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