# 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 $$\sum_{i=0}^\infty || y_{i}||^{2} \le \infty$$ & $$\sum_{i=0}^\infty || x_{i}||^{2} \le \infty$$ & $$s=\sum_{i=0}^\infty x_{i} \otimes y_{i}$$