Proposition 12.11.2.label Let $E, F$ be locally convex space over $K \in \RC$, then

where:

1. (1)

   The dual pairing between $E \otimes_{\pi} F$ and $L^{2}(E, F; K)$ is given by

   \[\angles{\sum_{k = 1}^n x_k \otimes y_k, \lambda}_{E \otimes_\pi F}= \sum_{k = 1}^{n} \lambda(x_{k}, y_{k})\]

   
   
2. (2)

   The dual pairing between $E \otimes_{\pi} F$ and $L(E; F^{*})$ is given by

   \[\angles{\sum_{k = 1}^n x_k \otimes y_k, T}_{E \otimes_\pi F}= \sum_{k = 1}^{n} \dpn{y_k, Tx_k}{F}\]

   
   
3. (3)

   The dual pairing between $E \otimes_{\pi} F$ and $L(F; E^{*})$ is given by

   \[\angles{\sum_{k = 1}^n x_k \otimes y_k, T}_{E \otimes_\pi F}= \sum_{k = 1}^{n} \dpn{x_k, Ty_k}{E}\]