Definition 13.12.4 (Injective Cross Seminorm).label Let $E, F$ be locally convex spaces over $K \in \RC$, $S \subset E^{*}$ and $T \subset F^{*}$ be equicontinuous, and $\lambda = \sum_{j = 1}^{n} x_{j} \otimes y_{j} \in E \otimes F$, then
\[[\lambda]_{S, T}= \braks{\sum_{j = 1}^n x_j \otimes y_j}_{S, T}= \sup_{\phi \in S, \psi \in T}\abs{\sum_{j = 1}^n \dpn{x_j, \phi}{E}\dpn{y_j, \psi}{F}}\]
is the injective cross seminorm of $\lambda$ with respect to $S$ and $T$. The family of all such norms induces the topology on $E \otimes_{\eps} F$. In particular, if $E, F$ are normed vector spaces, then
\[\norm{\lambda}_{E \otimes_\eps F}= \norm{\sum_{j = 1}^n x_j \otimes y_j}= \sup_{\substack{\phi \in E^* \\ \norm{\phi}_{E^*} \le 1}}\sup_{\substack{\psi \in E \\ \norm{\psi}_{F^*} \le 1}}\abs{\sum_{j = 1}^n \dpn{x_j, \phi}{E}\dpn{y_j, \psi}{F}}\]
is the injective cross norm on $E \otimes_{\eps} F$.
Post a Comment