Before my main question let me start with the following notions.
Let $X$ and $Y$ be locally convex spaces and let $T \colon X \rightarrow Y$ be a linear mapping. The adjoint of $T$ is an operator $T^{\prime} \colon D(T^{\prime}) \rightarrow X^{\star}$ where $D(T^{\prime}) = ( y^{\star} \in Y^{\star} \colon y^{\star}T \in X^{\star})$ given by $T^{\prime}y^{\star}= y^{\star}T$.
Let $X$ be an infra-barreled LCS and let $Y$ be a LCS. If we assume that $(X, \tau)$ is a K-space for some compatible topology (i.e. $\sigma(X, X^{\star}) \subset \tau \subset \beta(X, X^{\star})$) and $D(T^{\prime}) = Y^{\star}$ then $T$ is continuous with respect to the original topology of $X$.
(Reference: Theorem 3 in Endre Pap, Charles Swartz, A Locally Convex Version of Adjoint Theorem)
Definition 1
If $(X, \tau)$ is TVS, a sequence $(x_n)$ in $X$ is said to be a $\tau - K$-sequence if every subsequence of $(x_n)$ has a further subsequence $(x_{n_k})$ such that the subseries $\sum x_{n_k}$ is $\tau$-convergent to an element of $E$.
Definition 2
A topological vector space $(X, \tau)$ is said to be $K$-space if every sequence which converges to $0$ is a $\tau - K$-sequence.
Definition 3
A locally convex vector space $X$ is called infra-barreled if every bound absorbing barrel (i.e. a barrel absorbing all bounded sets in $X$) in $X$ is a neighborhood of $0$.
Question
I was wondering what should we assume on $X$ and $Y$ to get a continuous linear mapping $T$ when we have that $T^{\prime}$ is continuous (in its strong topology).
The problem is quite obvious when $X$ and $Y$ are normed spaces - so I'm not interested in this case.
In case of LCS: If $T$ is continuous then $T^{\prime}$ is also continuous (cos I assumed that $D(T)=X$), what about the converse?
Thank you in advance for any help and references.
BTW. I have a problem with { } brackets in LaTeX environment so I used ( ).
