Terminology of Operator Topologies Let $X,Y$ be normed vector spaces. In the case that $X,Y$ are Banach spaces, Folland defines the strong operator topology on the space of bounded linear maps $X \to Y$ to be the weak topology generated by the family of maps $T \mapsto Tx$ for each $x \in X$. Similarly he defines the weak operator topology to be generated by $T \mapsto f(Tx)$ where $f \in Y^*$, the space of bounded linear functionals and $x \in X$.
Is there any problem in vastly extending these definitions to define topologies on the space of all linear maps $X \to Y$ for arbitrary (not necessarily Banach) normed vector spaces? Would these be going against standard usage? It seems the restriction to bounded operators and Banach spaces is unnecessary. Is there anything wrong with this usage?
 A: The definition of the strong topology as that of pointwise convergence may be extended without issue to $\mathrm{End}(X)$, the space of linear maps $X\to X$. The restriction of this topology to the bounded linear maps $B(V)$ will then agree with the usual definition of the strong operator topology on $B(V)$.
The problem with doing this is that unbounded operators defined on all of $X$ are not the kind of unbounded operators you are interested in. Usually an unbounded operator $T$ is defined on a dense sub-space $D(T)$ of the Banach space $X$, and when talking about or defining this operator the domain is taken as part of its data.
This means that the space of those unbounded operators which one is usually interested in is not a reasonable operator space. You will face difficulties if you want to add or multiply such operators, simply because the domains of two operators you wish to add can be disjoint. Giving this space the "topology" of pointwise convergence is also problematic, because if the domains of the individual operators change you will not be able to consistently consider the convergence of $T_n x$ for a sequence $T_n$ and $x\in X$.
That said the notion of pointwise convergence is the usual way in which operations on unbounded operators are considered continuous. As an example in the measurable calculus defined by the spectral theorem, both the defining $f(T)=\int_{\sigma(T)}f(\lambda)\ dP(\lambda)$ and the domain of $f(T)$ are declared by pointwise considerations.
