J. Aust. Math. Soc.  76 (2004), 235-246
The exponential representation of holomorphic functions of uniformly bounded type

Thai Thuan Quang
  Department of Mathematics
  Pedagogical Institute of Quynhon
  170 An Duong Vuong, Quynhon, Binhdinh

It is shown that if $E$, $F$ are Fréchet spaces, $E \in (H_{ub})$, $F\in (DN)$ then $H(E, F) = H_{ub}(E, F)$ holds. Using this result we prove that a Fréchet space $E$ is nuclear and has the property $(H_{ub})$ if and only if every entire function on $E$ with values in a Fréchet space $F \in (DN)$ can be represented in the exponential form. Moreover, it is also shown that if $H(F^*)$ has a LAERS and $E\in (H_{ub})$ then $H(E \times F^*)$ has a LAERS, where $E$, $F$ are nuclear Fréchet spaces, $F^*$ has an absolute basis, and conversely, if $H(E \times F^*)$ has a LAERS and $F \in (DN)$ then $E \in (H_{ub})$.
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