Theorem 9.9.3. Let $E, F$ be metrisable locally convex spaces over $K \in \RC$, then for any $z \in E \td{\otimes}_{\pi} F$, there exists $\seq{\lambda_n}\subset K$ and $\seq{(x_j, y_j)}\subset E \times F$ such that:
$\sum_{n \in \natp}|\lambda_{n}| < \infty$.
$\limv{n}x_{n} = 0$ and $\limv{n}y_{n} = 0$.
$z = \sum_{n = 1}^{\infty} \lambda_{n} x_{n} \otimes y_{n}$.
Proof. Let $\seq{p_n}$ and $\seq{q_n}$ be increasing sequences of continuous seminorms that induce the topology on $E$ and $F$, respectively. For each $n \in \natp$, let $r_{n} = p_{n} \otimes q_{n}$, and $\td r_{n}$ be the continuous extension of $r_{n}$ to $E \td{\otimes}_{\pi} F$.
Let $u \in E \td{\otimes}_{\pi} F$, then there exists $\seq{u_n}\subset E \otimes_{\pi} F$ such that $\td r_{n}(u - u_{n}) < 2^{-n}/n^{2}$ for all $n \in \natp$. For each $N \in \natp$, let $v_{N} = u_{N+1}- u_{N}$, then
Since $r_{N} = p_{N} \otimes q_{N}$, there exists $\bracsn{(x_{N, k}, y_{N, k})}_{1}^{n_N}\subset X \times Y$ such that $v_{N} = \sum_{k = 1}^{n_N}x_{N, k}\otimes y_{N, k}$ and
By rescaling, assume without loss of generality that there exists $\bracsn{\lambda_{N, k}}_{1}^{n_N}$ such that
$v_{N} = \sum_{k = 1}^{n_N}\lambda_{N, k}x_{N, k}\otimes y_{N, k}$.
For each $1 \le k \le n_{N}$, $p_{N}(x_{N, k}), q_{N}(x_{N, k}) \le 1/M$.
$\sum_{k = 1}^{n_N}|\lambda_{k}| \le 2^{-N+2}$.
From here, let $\seqf{(x_j, y_j)}\subset X \times Y$ such that $u_{1} = \sum_{j = 1}^{n} x_{j} \otimes y_{j}$, then
where $x_{N, k}\to 0$ and $y_{N, k}\to 0$ as $N \to \infty$, and $\sum_{N \in \natp}\sum_{k = 1}^{n_N}|\lambda_{N, k}| < \infty$.$\square$