On the Space of Continuous Linear Operators on LCTVS Suppose that $X$ is a locally convex topological vector space (LCTVS) and that $L(X)$ denotes the space of all continuous linear operators on $X$.
Question. How can we construct a topology on $L(X)$ which compatible with the vector space structure of $L(X)$? 
I need help on this. Thanks in advance....

There is no natural topology on the dual space of a locally convex space unless the original space has a norm. In this case, if the original space is complete, then the $\sup_{B}$ norm is a norm on the dual space. Here $B$ is the unit ball in the original space. 
I think the conventional way to endow a topology on the dual space is as follows. Assume $H$ has a topology defined by a family of seminorms such that $\bigcap \{x:p(x)=0\}=\{0\}$. The subbase in this case is all sets of the form $\{x:p(x-x_{0})<\epsilon\}$. So a set $U$ in $H$ is open if and only if at every point $x_{0}\in H$, we have $p_{i},\epsilon_{i},i=\{1\cdots n\}$ such that $\bigcap^{n}_{j=1}\{x\in H:p_{i}(x-x_{0})<\epsilon_{i}\}\subset U$. 
Assuming this topology already well defined on $H$, we define a "weak-star topology" on $H^{*}$ by considering the family of seminorms given by $$\mathcal{P}=\{p_{x}:x\in H\}$$where $p_{x}(f)=f(x)$. Since $H^{*}$ is also a topological vector space, the above $p_{x}$ make it into a locally convex space in the same way as $H$. Some author claim that this is the only "natural" topology for the dual space of a locally convex space in the absence of a norm (See Conway). There is also a related personal note by Tao at here.
