Okay, what I'm trying to see is if are Banach spaces and is a net in the space of bounded operators such that is a Cauchy net in for all , then there exists such that in the strong operator topology.

This is what I have so far:

in the strong op. topology iff for all . Every Cauchy net in a Banach space converges. So we have that is a linear operator, but we have yet to prove (or disprove) that 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 are Hilbert spaces, then and by the Hellinger-Toeplitz theorem, since is everywhere defined and self-adjoint, we have that it's bounded. An analogous theorem gives us that is bounded and so is a bounded operator.

Edit: I don't think this previous paragraph works, since as we don't know that is bounded may not exist, or is there a way to define a formal adjoint of a linear operator with the properties ?

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