Dual of the linear map I know the meaning of dual of linear map in inner product spaces, also it is defined in Banach space[ Rudin functional analysis].
What is the definition of Dual of linear map if vector space are Frechet space/ or more generally locally convex sapce which is not normable.
 A: Definition of dual linear map in topological vector spaces over field $F$ is the same as in normable case. Given linear map $T:V\to W$ between topological vector spaces we define its adjoint as
$$
T' : W'\to V':f\mapsto f\circ T
$$
Motivation for this definition of dual linear operator from categorical point of view is the following. If you want to study some object (for our purposes linear operators between topological vector spaces) its useful to study maps from this object ($\mathrm{Hom}(V,W)$ in our case) into something simple (field of scalras $F$ will fit). This useful maps can be obtained by applying contravariant $\mathrm{Hom}(-,F)$ functor to the linear operators $T\in\mathrm{Hom}(V,W)$. Then you get $\mathrm{Hom}(T,F)=T'$
Unfortunately, this definition may not have much sense, because for general topological vector spaces its dual can be trivial. But if you restrict your interest to locally convex spaces this problem won't rise.
A: Let $X$ and $Y$ two locally convex spaces, and $X'$, $Y'$ their respective duals. If $T\colon X\to Y$ is linear, we can define $T^*\colon Y'\to X'$ by 
$$\forall  l\in Y', \forall x\in X, \quad \langle T^*l,x\rangle_{X',X}=\langle l,Tx\rangle_{Y',Y}.$$  
