Proposition 12.13.8.label Let $\seqi{E}$ be nuclear spaces over $K \in \RC$, then $\prod_{i \in I}E_{i}$ is nuclear.
Proof, [Theorem III.7.4, SW99]. Let $F$ be a Banach space and $T \in L(\prod_{i \in I}E_{i}; F)$, then there exists $J \subset I$ finite and $\wh T \in L(\prod_{j \in J}E_{j}; F)$ such that the following diagram commutes:
\[\xymatrix{ \prod_{i \in I} E_i \ar@{->}[r]^{T} \ar@{->}[d]_{\pi_J} & F \\ \prod_{j \in J}E_j \ar@{->}[ru]_{\widehat T} & }\]
By Proposition 12.8.3, $\prod_{j \in J}E_{j} = \bigoplus_{j \in J}E_{j}$. By Proposition 12.13.7, $\bigoplus_{j \in J}E_{j}$ is a nuclear space, so $\widehat T: \bigoplus_{j \in J}E_{j} \to F$ is a nuclear operator. As the composition of a continuous operator and a nuclear operator, $T$ is nuclear by Proposition 12.12.4. Therefore $\prod_{i \in I}E_{i}$ is a nuclear space.$\square$
Post a Comment