Let $X$ and $Y$ be separable Banach spaces and $L(X,Y) $ be the Banach space of bounded linear operators from $X$ to $Y$. Suppose $A$ is a norm closed finite codimensional subspace of $L(X,Y)$.

My question is: For which spaces $X$ and $Y$ is $A$ (finite codimensional) Borel in the strong operator topology?

Recall that the strong operator topology is the topology given by the pointwise convergence of nets of operators.

Edit: A previous iteration of this question asked for spaces $X$ and $Y$ so that $A$ closed in the strong operator topology.

`$X^*$`

need not be weak$^*$ closed--consider the kernel of a functional in`$X^{**}\sim X$`

. – Bill Johnson Apr 6 '13 at 1:20