Sten Bjon

A general approach to infinite-dimensional holomorphy

In this paper we present a general theory for holomorphic functions which is based on continuous convergence instead of topologies. The theory can be applied to locally convex spaces and bornological spaces.

Homomorphisms on some function algebras

In this note we prove, for some classes of real locally convex spacesE including all complete Schwartz spaces, that every non-zero homomorphism on the algebraC∞ (E) ofC∞-functions onE is given by a point evaluation at some point ofE.

Function algebras on which homomorphisms are point evaluations on sequences

In the study of the spectrum of a subalgebraA ofC(X), whereX is a completely regular Hausdorff space, a key question is, whether each homomorphism ϕ:A→R has the point evaluation property for sequences inA, that is whether, for each sequence (fn) inA, there exists a pointa inX such that ϕ(fn)=fn(a) for alln. In this paper it is proved that all algebras, which are closed under composition with functions inC∞ (R) and have a certain local property, have the point evaluation property for sequences. Such algebras are, for instance, the spaceCm(E) (m=0,1,...,∞) ofCm-functions on any real locally convex spaceE. This result yields in a trivial manner that each homomorphism ϕ onA is a point evaluation, ifX is Lindelöf or ifA contains a sequence which separates points inX. Further, also a well known result as well as some new ones are obtained as a consequence of the main theorem.

CHARACTERIZATION OF SCHWARTZ SPACES BY THEIR HOLOMORPHIC DUALS

Let U be an open subset of a locally convex space E, and let Hc (U, F) denote the vector space of holomorphic functions into a locally convex space F, endowed with continuous convergence. It is shown that if F is a semi- Montel space, then the bounded subsets of HC{U,F) are relatively compact. Further it is shown that JE is a Schwartz space iff the continuous convergence structure on the algebra Ft(U) of scalar-valued holomorphic functions on {/, coincides with local uniform convergence. Using this, an example of a nuclear Frechet space E is given, such that the locally convex topology associated with continuous convergence on H(E) is strictly finer than the compact open topology. Thus, the behavior of the space HC(E) differs in this respect from that of its subspace LCE of linear forms and that of its superspace CC(E) of continuous functions. Introduction. In (11) H. Jarchow has proved that a locally convex space (les) E is Schwartz if and only if continuous convergence and local uniform convergence coincide on the dual of E. It is natural to ask if there is a holomorphic analogue of this result: Is a space E Schwartz if and only if continuous convergence and local uniform convergence coincide on a space H(U) of holomorphic functions on some open subset U of El We give a positive answer to this question using a result, closely related to Jarchows, for continuous m-homogeneous polynomials (5). Further we prove that the space H(U) has the Montel property (bounded sets are relatively compact) when endowed with continuous convergence. For spaces of linear forms (on locally convex spaces) as well as for spaces of con- tinuous functions (on e.g. completely regular spaces) it is known that the locally convex topology associated with continuous convergence is the compact-open topol- ogy (cf. (1 and 10)). Using the above holomorphic characterization of Schwartz spaces, we provide an example which shows that spaces of holomorphic functions behave quite differently: There exists a nuclear Frechet space E, such that the locally convex topology associated with continuous convergence on H(E) is strictly finer than the compact-open topology. We recall some notation and definitions. All vector spaces in this paper are complex. A function f:U—*F into a convergence vector space (cvs) F is Gâteaux- holomorphic if the function A t—> I o f(x + Xh) is holomorphic in a neighborhood of zero for each x G U, h G E, and l G LF. It is holomorphic if it is Gâteaux- holomorphic and continuous. Let H(U, F) be the vector space of holomorphic

Algebras of Holomorphic Functions

1 Let H(U) denote the algebra of holomorphic functions on an open sub- set U of a complex locally convex Hausdorf space E and let H,(U) and H,(U) denote this algebra when supplied with the continuous and the associated equable convergence structures, respectively [3,4]. One objec- tive of this note is to show that the convergence algebras H,(U) and H,(U) in some (quite general) cases contain total information concerning the space U in the sense that the spectrum of H,(U) or H,(U) is homeo- morphic to U, when it is given the continuous convergence structure. The finest locally convex topologies on H(U) coarser than the con- vergence structures of H,(U) and H,(U) will be denoted by K and A. In [lS] H. Jarchow treats analogously defined topologies y and q on spaces of continuous linear forms. In [S] we proved that H,(U) coincides with H,,(U) (cf. [19]) if U is a Lindelof space and that il is always liner than the ported topology z, (cf. [9]). Isidoro [ 141 and Mujica [ 181 have characterized the spectrum (as a set) of H,,(U), Hro( U), and H,,(U) and it has been used, e.g., for constructing the envelope of holomorphy of U. Using the result about the spectrum of H,(U) and H,(U), mentioned above, we obtain information on the spectrum of H,(U) and H,(U). Finally we prove that the bornological topology associated with the con- tinuous convergence structure on the vector space H( U, F), of holomorphic functions U + F, coincides with the compact-open topology, when U is an open subset of a DFM-space E and F is a metrizable locally convex space. This result is analogous to a similar result of B. Miiller [19] concerning spaces of continuous linear functions. For topological spaces X and Y let C,(X, Y) be the set of continuous functions X + Y endowed with the continuous convergence structure, i.e., the coarsest convergence structure for which w: C(X, Y) x X-+ Y, o(f, x) =f(x) is continuous. If E is a locally convex space, then C,(X, E) is 207

On an integral transform by R. S. Phillips

AbstractThe properties of a transformation

On a Bornological Structure in Infinite‐Dimensional Holomorphy

f \mapsto \tilde f_h

On nuclear limit vector spaces

by R.S. Phillips, which transforms an exponentially bounded C0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete Lc-embedded space E is closed under the transformation. It is shown that