By N. J. Kalton, N. T. Peck, J. W. Roberts
This e-book provides a thought inspired via the areas LP, zero ≤ p < l. those areas will not be in the community convex, so the equipment often encountered in linear research (particularly the Hahn-Banach theorem) don't observe right here. questions on the scale of the twin area are in particular vital within the non-locally convex surroundings, and are a critical subject matter. numerous of the classical difficulties within the region were settled within the final decade, and a few their options are awarded right here. The e-book starts with concrete examples (lp, LP, L0, HP) ahead of happening to common effects and critical counterexamples. An F-space sampler should be of curiosity to investigate mathematicians and graduate scholars in sensible research.
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Additional info for An F-space Sampler
L outer Hp spaces if 17 exp(l/2n [0 Zn eit+z logW(t)dt) it e V 6 LF and z y e R. Then the Canonical Factorization Theorem (Duren, p. 6) associated to 8(2) = H Ianl/a is n is also an inner Duren, p. 19). If u S = BSl 26D 1 if an = 0). an (cf. S inner function and takes the form = exp( J0 (eit+z/eit z) on du(t)) (with respect to ( n,n]. 7) I; function and has is called a singular If so that < w function (where function. may be further factorized. (possibly finite) E(l la B inner P The inner function (a ) n S where B S can be factored in the is a Blaschke product and S1 is a singular inner function.
1 1 c_ 1 > 0 and m Eci<1+c i=0 where C = suan . 1 This, I 1 P . 9) completes the proof is contained in Hp. U is an example of a reproducing kernel and can be used to produce a projection of Ll(#) onto Bp where d is the measure on O = (l lwl) dx. D given by _46_ In fact if f e Ll(u) Pf(z) Then P we define Pf e Bp by = ID K(z,w)f(w)dx(w) = JD J is a projection of w (z)f(w)dn(w). Ll(u) Since the unit ball of analytic functions on D, onto its subpsace Bp Bp is a normal family of it is not difficult to see that is isomorphic to a dual Banach space.
Thus we can find see Duren p. ) p = (l la I2)l/pf(a ) n is an open mapping of and : Hp s 1 T Then the map k. for every ST is bounded and subspace isomorphic to 1 . p (This is a deep result - gn e Hp with is defined by tngn l is a projection of onto a Hp is not 1p. 4. Proof. HD D In view of this we next establish that isomorphic to supngnlp < co contains a subspace isomorphic to In fact by a theorem of Paley (Duren, p. 1?. e. inf n k k+l /n k > 1. Let us give a simple proof for the special case nk = 3 k .
An F-space Sampler by N. J. Kalton, N. T. Peck, J. W. Roberts