Okay, what I'm trying to see is if $\displaystyle X, Y$ are Banach spaces and $\displaystyle (T_a) \subset \mathcal{B} (X,Y)$ is a net in the space of bounded operators such that $\displaystyle (T_ax)$ is a Cauchy net in $\displaystyle Y$ for all $\displaystyle x\in X$, then there exists $\displaystyle T\in \mathcal{B} (X,Y)$ such that $\displaystyle T_a \rightarrow T$ in the strong operator topology.

This is what I have so far:

$\displaystyle T_a \rightarrow T$ in the strong op. topology iff $\displaystyle T_ax \rightarrow Tx$ for all $\displaystyle x\in X$. Every Cauchy net in a Banach space converges. So we have that $\displaystyle Tx:= \lim_{a} T_ax$ is a linear operator, but we have yet to prove (or disprove) that $\displaystyle T$ is bounded. The case where the net is a sequence follows immediately from the Banach-Steinhaus theorem because every convergent sequence in bounded, but this need not happen for an arbitrary net.

Let's assume for a moment that $\displaystyle X, Y$ are Hilbert spaces, then $\displaystyle T= \frac{T+T^*}{2} +\frac{T-T^*}{2} =T_1+T_2$ and by the Hellinger-Toeplitz theorem, since $\displaystyle T_1$ is everywhere defined and self-adjoint, we have that it's bounded. An analogous theorem gives us that $\displaystyle T_2$ is bounded and so $\displaystyle T$ is a bounded operator.

Edit: I don't think this previous paragraph works, since as we don't know that $\displaystyle T$ is bounded $\displaystyle T^*$ may not exist, or is there a way to define a formal adjoint of a linear operator with the properties $\displaystyle \langle x,Ty \rangle = \langle T^*x,y\rangle$?

Now, I'm at a loss as to how to approach the general case...