Let H be a Hilbert space, Mc H a convex subset, and (x₁) a sequence in M such that ||x₁|d, where d = inf ||x||. Show that (x,) converges XEM in H. Give an illustrative example in R² or R³.

Q: 1.3 State and prove the Cauchy Criterion for the sequence of real-valued functions, with R equipped…

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Q: n 11. Let X be a non-empty set that is bounded from below. Show that there is a decreasing sequence…

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Q: 5. Prove that the set of points at which sequence ef measurable real-valued functions converges (to…

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Q: For an orthonormal sequence {u} in a Hilbert space H, prove the equivalence of the conditions (i)…

A: Given:  is orthonormal sequence in a Hilbert space .(i)  for all  implies .(ii)  for all .To prove:…

Q: Let (v1,..., vk) and (w1,..., wi) be linearly independent sequences wi) is linearly independent if…

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Q: 2. Construct a sequence of unbounded functions In : [0,1] → R that converges pointwise to a bounded…

A: Step 1:

Q: Prove that if {xn} is a sequence such that 1+ n |æn| < for all n E N, 1+ n2 then the sequence {xn}…

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Q: Assume A is a bounded, non-empty subset of R. Let r = sup(A). Prove there is a sequence from A…

A: Given r=sup(A). Let r∈A.  Consider the sequence {xn} such that xn=r ∀n∈ℕ.Then {xn} is a sequence…

Q: Let 1 ≤ p < ∞ and let {fn} be a sequence of functions in Lp[a, b]. Suppose there is a function f ∈…

A: We are given a sequence of functions {fn} in Lp[a,b].

Q: (10) Let H be a Hilbert space and let {an} be a sequence in H. We say that {an} converges weakly to…

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Q: Let {ai} be a sequence in a metric space X,d. Prove that lim an = p if and only if lim d(an, p) =…

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Q: A Cauchy sequence f} in L₂ converges to an element f in L₂

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Q: Does the Bounded Convergence Theorem hold if m(E)<∞ but we drop the assumption that the sequence…

A: Given below clear explanation

Q: 5 Let (sn) be a bounded sequence, and let k be a non number. onnegative real (a) Prove lim sup(ksn)…

A: Let Sn be a bounded sequence and let K be a non-negative real no. (a)  Let limn→∞Sup Sn=l Let K&gt;0…

Q: Let (an) be a sequence in which every term is from some given finite set R. Prove that (an) has a…

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Q: a. In a metric Space (X, d) if (xn) and (yn) are two sequences in X such that : X - x€ X and yn → y…

A: a) In a metric space (X,d) Let (xn) and (yn) be two sequences such that xn -&gt; x and yn -&gt; y…

Q: Let f be a uniformly continuous function on a set E. Prove that for any Cauchy sequence {n} CE, the…

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Q: If v = {[(xn)]; (xn) is a cauchy sequence in V } Show that (ỹ, ||. || ) is a complete normed space…

A: The question states: "If v = {[(xn)]; (xn) is a Cauchy sequence in V}, show that (~, . ||) is a…

Q: Find limits of the following sequences or prove that they are divergent. (a) an =√n(−1)^n

A: The given sequence is an=n-1n We will use the result that for any sequence an such that if the…

Q: i) Let {n} a sequence of non-negative terms that is monotone decreasing and în → 0. If SnX1 X2 + x3…

A: Given that, { xn } is a sequence of non negative terms and this sequence is monotonic decreasing,…

Q: There is a countable collection of compact sets whose union is not compact. Select one: O True False

A: NOTE: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Generalization of Exercise 2.3.8. Suppose {xn}~_=1 and {yn}_₁ are two bounded sequences of real…

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Q: A. Let (X, d) be a metric space. Define a diameter of a subset A of X and then the diameter of an…

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Q: 1.57 Let r, and yn be any two bounded sequences of real numbers. Prove that lim sup(xn + Yn) < lim…

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Q: Fend the Rmit of the following aq =2 %3D a2=12-52 %3D 2-2+2+2+ ... + U.

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Q: Assume f: R→→ R. and for some c < 1 we have for all x, y € R, |f(x) = f(y)| ≤ c|x − y|. (a) Prove…

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Q: 2) Prove that rxz(1) <rzz(0), where r denotes the autocorrelation function of a sequence r(n) and I…

A: Autorrelation function is mathematically given by, rxx(l) = E[X(t+l).X(t)] , for all t. Similarly,…

Q: is a bounded sequence in H and since A is a compact operator, there is a subsequence (Yn₁) such that…

A: We need to explain this portion of the proof: