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L^{1}(H) $\iff$ $s=\sum_{i=0}^\infty x_{i} \otimes y_{i} $

$s\in L^{1}(H)$ $\iff$ $ \exists \{ x_{i} \} , \{ y_{i} \} \subseteq H $ s.t $ \sum_{i=0}^\infty || y_{i}||^{2} \le \infty$ & $\sum_{i=0}^\infty || x_{i}||^{2} \le \infty $ & $s=\sum_{i=0}^\infty x_{i} \otimes y_{i} $