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Functional Analysis, Holomorphy and Approximation Theory: by Jorge Alberto Barroso

By Jorge Alberto Barroso

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Ology has a base of 0-neighborhoods which are closed in the weaker complete topology of SUMMERS ha. 6 ( (A1)’ ,Al) from which we obtain completeness Finally, at this point, (f) is obvious. 3 a matrix I = guished. (e), we would like to make the following observation. 9. Remark. (3)]), By a well known theorem of Grothendieck [4] (see also the strong dual of a metrizable space is bornological if, and only if, it is barrelled. c. space E actually coincide. To see this, we utilize [3, Proposition 31 where a suitable form of Grothendieckts main lemma is derived, taking T1 and the bornological and barrelled topologies associated with respectively, and complete whereby T3 = u(E’,E).

Ph. D. Dissertation. UCLA 1978. 7. D. SARASON, Function Theory on the unit circle. Lecture Notes Virginia Pol. Inst. and State Univ. Virginia 1978. Added in Proof After submitting this paper the authors develloped considerable the subject and summarized their results in the paper "Generalized Toeplitz kernels, Hankel forms and Sarason's commutation theorem", to appear in Acta Cientifica Venezolana. In particular, R. Arocena proved the conjecture of Section 5 concerning the extension of the Adamjan-Arov-Krein parametrization to 3(M), for M > 0.

Of all finite subsets of and let If, for each ~~, yJ = ( ~ ~ ( i ) ) ~ where yJ(i) = y(i) elsewhere, then clearly yJ E B fl q(I), y defined by = (z(i))iEI, and to be zero elsewhere, is an ele- But since To finish the proof, take to 2; 47 pointwise on and noting that convergence on I. 5(I) denote the system J E 3(I), i E J for we put and is zero and Applying the fact (’J, JE5 (I) that B n q(1) converges c v ? 4,p(I)1 is closed in the topology of pointwise I, we obtain y E ? , B c ? L (I)1 as P Lp(I)l is also possible was desired.

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