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By N. J. Kalton, N. T. Peck, J. W. Roberts

This booklet provides a thought prompted through the areas LP, zero ≤ p < l. those areas are usually not in the community convex, so the tools frequently encountered in linear research (particularly the Hahn-Banach theorem) don't follow the following. questions on the dimensions of the twin area are in particular very important within the non-locally convex atmosphere, and are a valuable subject. numerous of the classical difficulties within the zone were settled within the final decade, and a few their suggestions are provided the following. The booklet starts off with concrete examples (lp, LP, L0, HP) prior to occurring to normal effects and significant counterexamples. An F-space sampler might be of curiosity to analyze mathematicians and graduate scholars in useful research.

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M Let E n. be a closed subset of T, with is normalized Haar measure on the circle dm = (2n) ld9). e. the set of probability A(E) the set of u e M1(E) (in where II of the form w-dm p e Lw(m). 12) . inf *11<9<'n I(SM (rn k Let us suppose have been chosen. ,Ek, :1 . l g j s k. ,nk Mk 6 A(Ek) inf 25D IS n (2)! k > 0 and #1,.

Vilp < l Lp(T). zNwi e Hp and 1 for l < i < n. e. H is not locally convex. 3 to z P Hp H9 has a separating L . P contains a complemented subspace isomorphic . Proof. for some Let (ak) 5 > 0, be a uniformly separated sequence in D i. e. -38- la H j=1 k aj/ 1 3 jak I > a jsk (Tf) n If Tgn = en. onto HD S 1p e Hp : S(t) E h then S defined by n £9. 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 .

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. Bp Now a theorem due to Lewis and Stegall (1973) asserts that a complemented subspace of which is a dual Banach space is isomorphic to 11. 9. Since Bp Bp is isomorphic to has an unconditional basis, it was a natural question to ask whether of course, basis for 11 H9 has an (unconditional) basis.

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