Adjusted citation formats. Moved citation off of named theorems if possible.
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Bokuan Li
2026-03-19 23:58:16 -04:00
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\begin{definition}[Cross Seminorm, {{\cite[III.6.3]{SchaeferWolff}}}]
\begin{definition}[Cross Seminorm]
\label{definition:cross-seminorm}
Let $E, F$ be locally convex spaces over $K \in \RC$. For any convex circled sets $U \in \cn_E(0)$ and $V \in \cn_F(0)$, let $p: E \to [0, \infty)$ and $q: F \to [0, \infty)$ be their \hyperref[gauges]{definition:gauge}. For any $z \in E \otimes_\pi F$, let
\[
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\end{enumerate}
\end{definition}
\begin{proof}
\begin{proof}[Proof {{\cite[III.6.3]{SchaeferWolff}}}. ]
(1): Let $\lambda \in K$, then for any $\seqf{(x_j,y_j)} \subset E \times F$,
\[
|\lambda| \sum_{j = 1}^n p(x_j)q(y_j) = \sum_{j = 1}^n p(\lambda x_j)q(y_j)