Problem 1 decide if each of the following claims is true or

Question

Problem 1 Decide if each of the following claims is true or false. Justify your answer with a brief argument (for example a reference to the textbook, including exercises done in class, a short proof, or a counterexample

(i)  CEO, 1] is a Banach space when equipped with the norm 111112 = (f_o^l If (x)I² dx) .

(ii)   co, is a Banach space when equipped with the norm 11116 = suNcom If (x)I.

(iii)  If M is a closed subspace of a Hilbert space H and if M¹ = {O}, then M = H.

(iv) If H is a Hilbert space with a complete orthonormal set (ca):^°=1, then at, le„ defines an element in H.

(v)  If f E     al has Fourier coefficients c„(f),n EZ, then E°°,₂__. le_n(f)1² < 00.

(vi) Every linear operator T: E F between two normed spaces E and F is bounded.

(vii) Every continuous linear operator T: E —> F between two normed spaces E and F is bounded.

Problem 2  Consider the vector space C[0,1] equipped with the inner product (Ls) = f (x)9(x) dr,   f,s E C[0,Jol

and associated norm 11111 = (1, f)'¹². Let M be the subspace of C[0,11 spanned by the functions th(x) = 1 and tz2(x) = x. Let h E C[0,1] be given by h(x) = f.

=      I (x)9(x)ax^,           1,9 e t.4u,

and associated norm 11111 = (f, f)'/². Let M be the subspace of C[0,1] spanned by the functions u_i(x) = 1 and u₂(x) = x. Let h E C[0,1] be given by h(x) = f.

(1) Find an orthonormal basis for M.

(ii) Find a, b E C which minimize the norm IIh — au₁ — bull'. Sketch the graphs of h and au' + bu2 in the same coordinate system.

(iii)Determine IIhlI, Ilh — aui — 611211, and Haul + WA, where a, b are as in question (ii). Problem 3 (16 points). Consider the Hilbert space L²(--7r, 7r) with its usual inner product (f, 9) = f (x)g(x)dx,    f, g E L²(—/r,

Let (e„)°?__₀₃ be the orthonormal basis for L²(—ir , Tr) consisting of the trigonometric func­tions

     e_i,(x) = 2T e^l",       x EV

and let c„(f) = (f,e„) denote the Fourier coefficients of f E

Consider the functions f,g:               C given by f(x) = IxI and g(x) = sin(x). You

may use that sin(x) = (e — e^-lx)/2i, and you are not asked to prove this fact.

(1) Calculate 11111² and 11911².

(ii)      Determine the Fourier coefficients c„(f) and c„(g) for all n E Z.

(iii)   Determine act__coc_n(f)c_n(g).

Problem 4 (8 points). Consider the Hilbert space L²(-1,1) and the linear functional F on L²(-1, 1) given by1

F(f) = ji_lt f _(t) dt,          f E

Show that F is bounded and determine IIFII• Find f E L²(-1,1) such that Ilf II = 1 and IF(f)I=

Problem 5  Consider the Hilbert space e2 and consider the linear operator .9: ea e given by

S(Xj, X2₎ X3, X4) X5, X61 • • • ) = (X2) XII X4) X3) XIS) X5> • • • ))    €²,

1Sx = y . (_yn)_,00 _{1,                       y.} . xn+i, n odd

x„_1, n even.

You are given, and you need not prove, the facts that S is linear and that Sx E €² for all x E /2.

(i)    Show that S E 2(12) and determine 11Sii•

(ii)    Show that Ss = S and that S² = I.

(iii)    Put P = i(I + 5). Show that P is a projection, i.e., show that P = P and that P = P². [Hint: You may use, without giving a proof, that I is Hermitian, i.e., I' = I. You may also use that the identity operator I acts as a multiplicative unit on 2(9), i.e., IT = TI = T for all T E sew)"

(iv)    Put x= (1,2,3,4,3,2,1,0,0,0,                ) E e.

Determine the vectors Px and x — Px and calculate the norms 114 IIPxII, and Ilx — Prd.

Problem 6 (10 points). Let 4 be the subspace of C² consisting of all sequences x (x.):°_,₁ which eventually are zero, ef. Example 2.7 in the textbook. Equip C² and 4 with the norm Hall = (x, x)¹/² arising from the usual inner product on e.

Consider the linear functionals F,G: 4 —) C given by

2011VW= E x,„ G(x) = E x., x = (xa)wl E 4.

.m1    nalShow that G is bounded and that F is unbounded. Determine 11011.A comment about Fourier coe_fficients: Fourier coefficients for a function f E L²(—sr,rr) are at the lectures defined to be c„(f) = (f , e_n), where e„(x) = (21r)^-112einz, x E I-7T, srbwhence f  en(nen. The textbook uses Fourier coefficients denoted e_n, whichsatisfy f(x) Ec°  c_nenz. The two notions of Fourier coefficients are related via theformula: c„ = (2x)^-1/²e„(f). In this exam, Fourier coe_fficients will always be of the first kind, that is c_alf) = (f,e„).

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