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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.

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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 $y$.

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