9. If X is finite dimensional Hilbert space, then it has Orthonormal basis.10. Let {x} be an Orthonormal sequence in a pre-Hilbert space X and let xe X. Show thatx=y LM, where y = £4,5, and M =[Full-11. Show that In a finite dimensional normed space, each closed and bounded set is compact.12. Let A be a subset of a Hilbert space X . Show thata.. A¹ = A¹b.. A¹ = [4]c.. A is dense in X iff A² = {0}13. Let X,Y are Hilbert space on a field F. Show that X,Y are Hilbert Isomorphic iffdim X = dim Y

As per the guidelines I have solved first question.

Answered: . If X is finite dimensional Hilbert…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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6. Let (en) be a complete orthonormal sequence in a Hilbert space H and let (2n) be a sequence of scalars. = (a) Show that there exists a unique operator T on H such that Ten= λnen. (b) Show that T is bounded if and only if the sequence (λn) is bounded. (c) For a bounded sequence (λn), find the norm of T.
5
3. (a) Consider the function defined over three binary variables: f (x1, x2, 3) = (x₁^₂^x3). We aim to find a such that, for any x= [1, 2, 3], where x; = {0, 1}: 0x +000 when f (x1, 2, 3) = 1, and 0x+000 when f (x1, x2, 3) = 0. = If 000 (no offset), would it be possible to learn such a 0? Yes No Would it be possible to learn the pair and 00? Yes
(c) Let un be the sequence in 1² with 1 in the nth place and zeros elsewhere. Prove that the set {u} is an orthonormal basis for 1²
linear algebra
X vector space ||.|| and ||.||' if arbitrary norms are equivalent According to ||. || ' norm Cauchy sequence ||. || norm is also the Cauchy sequence. Research and Prove
Simplify the Boolean function F together with the don't-care conditions d in (1) sum-of-products form and (2) product-of-sums form. F (w, x, y, 2) =Σ 0, 1,23, 7 8, 10)
Provide a coherent answer for items C and D.
Q1\ A\ Define and give an examples with its solutions for the following terms: (1) Strongly converges (2) Orthogonal set (3) Pre-Hilbert space

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. If X is finite dimensional Hilbert space, then it has Orthonormal basis. 0. Let {x} be an Orthonormal sequence in a pre-Hilbert spaceX and let xe X. Show that