# Why does the series converge in the space of bounded linear operators?

A Banach space $$E$$ is given, and so are sequences $$(u_i)$$ and $$(v_i)$$ in $$E$$ and $$E'$$, respectively. These sequences satisfy $$\sum\left|\langle x,v_i \rangle\right|\left|\langle u_i, y \rangle \right|\leq 1$$ for every $$x$$ and $$y$$ in the unit balls centred at the origin of $$B$$ and $$B'$$, respectively. Why does $$T(x)=\sum v_i \otimes u_i(x)=\sum \langle x, v_i\rangle u_i$$ converge for all $$x\in E$$?

I have been stuck on this question for a while, and I would appreciate any help.

## 1 Answer

This follows from Cauchy's criterion. In fact, if $$T_k$$ denotes the partial sum $$\sum_{i=1}^kv_i\otimes u_i$$, then $$\left|\langle(T_m-T_n)x,y\rangle \right|$$ is small, whenever $$x$$ and $$y$$ lie in said balls and both $$m$$ and $$n$$ are large enough. It then suffices to take the supremum over all relevant $$y$$.