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Problem 1. We denote by the set of all sequences (UK)x=1,2,... = (U1, U2, ...) (ux E C) u= satisfying luxl <00. Moreover, we
Definition For A CH(A + o), we define A= {U € Hiu I A}. Theorem Ais a closed subspace of H. Proof. Let u, u E A. For any v E


Moreover, set f(0) = 62. Write the Fourier expansion off with respect to the system of trigonometric functions in L(-, 7). P

Sorry please neglect the bottom picture
which says "moreover ...".
I am happy to upbote if you solve (1)-(5).
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• 2 = { us (uus. buce: lex lo conoy llighet u = (h, us, ...), 82 (0, 2, ...) E d², ;, ;6C Then I tugel (& and I lovalca ha le+ altv) tw tu, u, we? i et u (ay, uz, I E d2 ; uitd. Then (unuz, ...)+(0,0,...) = (1+0, uz to, - - -) (m, uz, .--) ituito. Ual y Elcuxľ ( DC K21 -) 66e2 & CAC, ut l2. & eod.tc, ure2 ay (cm, curi: sy o cu el? vii) cet c, d ec, ufe². c(du) (da, , duz,(cur, cur, .-) + ( day, deez, .-) clu, uz, - -) + dlu, uz, .-) 2 cut du v cid el, url2 *) tEc is identity Then for all unlu,unet (and be any Cauchy sequence in e?, where am (s ....). Then for every Exo 7 N&t. & in, n7N d(am, an) = 11am-xull 12 = Lam-we get 21 s (m) ; 1² c. er J에 Y (m) hij El 2 completeness of e2 let kya then for m YN an-x - die since ant l2 , it follows by

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