PRELIMIN ARIES AND NOTATION for all uE H.( We only consider separable Hilbert spaces, i.e., spaces in which orthonormal bases are countable ) Examples of orthonormal bases'are the Her mite functions in L(R), the sequences en defined by (en),=8n,,, with n,jE Z in e (Z)(i. e, all entries but the nth vanish), or the k vectors el,.,ek in C defined by( ec)m= &L, m, with 1 <l, m<k(We use Kronecker's symbol 6 with the usual meaning:6,=1ⅱfi=j,0ft≠ A standard inequality in a hilbert space is the Cauchy-Schwarz inequality (v,t)≤ lull wll, (0.0.6) easily proved by writing(0.0.5)for appropriate linear combinations of v and w forf,g∈L2(R 1/2 dxlg(x川 c=(cn)n∈z,d=(dn)n∈z∈2(z), 1/2 ∑a≤(∑a of(00.6) (0.0.7) H are linear from H to another Hilbert often咒 Itself. Explicitly, if A is an operator on H, then A(λ1a1+入2u2)=A1Au1+A2A An operator is continuous if Au- Av can be made arbitrarily small by making u-v small. Explicitly, for all e>0 there should exist &( depending on e)so that‖l-ⅶ≤ 8 implies‖Aa-A啡≤∈. If we take v=0,e=1, then we find that, for some b>0,‖Al≤1if|≤b. For any w∈ we can define clearly/l≤ b and therefore Awll=ll‖Anln≤b-l.fr I Aw/lull(w#0)is bounded, then the operator A is called bounded. We have just seen that any continuous operator is bounded; the reverse is also true. The norm‖A‖ of A is defined by ‖A= sup Aull/ll=sup‖Au‖l (0.0.8) t∈,‖u‖10 It immediately follows that, for all uE 7 ‖Atll≤‖Alll Operators from H td c are called "linear functionals. " For bounded linear functionals one has Riesz@representation theorem: for any &: H-C, linear and