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If $(X,\tau)$ is a topological vector space, a sequence $\{x_k\}$ in X is said to be a $\tau$-$K$ sequence if every subsequence of $\{x_k\}$ has a further subsequence $\{x_{n_k}\}$ such that the subseries $\sum_kx_{n_k}$ is $\tau$-convergent to an element of $X$.

A topological vector space $(X,\tau)$ is said to be a $K$-space if every sequence which converges to $0$ is a $\tau$-$K$ sequence. A subset $A$ of $X$ is said to be $\tau$-$K$ bounded if for every sequence $\{x_n\}\subseteq A$ and every scalar sequence $\{t_n\}$ converging to $0$, the sequence $\{t_nx_n\}$ is a $\tau$-$K$ sequence.

Let $E$ and $F$ be Hausdorff locally convex topological vector spaces (cf. also Locally convex space; Hausdorff space) and let $T:E\to F$ be a linear mapping. The domain of the adjoint operator, $T'$, is defined to be

\begin{equation}D(T')=\{y'\in F':y'T\in E'\}\end{equation}

and $T':D(T')\to E'$ is defined by $T'y'=y'T$.

The first adjoint theorem was proved by E. Pap [a3] for operators defined on pre-Hilbert $K$-spaces. There exists a pre-Hilbert $K$-space which is not a Hilbert space, [a2]. A generalization of the adjoint theorem for normed spaces was given in [a1], [a4]. It reads as follows.

Let $E$ be a normed $K$-space, let $F$ be a normed space and let $T:E\to F$ be a linear operator. Then the adjoint operator $T'$ is a bounded linear operator on $D(T')$.

In the proofs of all these theorems, so-called diagonal theorems were used (cf. also Diagonal theorem). As a simple consequence, a proof of the closed-graph theorem without the Baire category argumentation was obtained, [a3], [a4], [a6], [a8], [a9].

There is a locally convex generalization of the adjoint theorem [a5], [a7]: $T'$ is sequentially continuous with respect to the relative $\sigma(F',F)$- (weak) topology on $D(T')$ and the topology on $E'$ of uniform convergence on $\sigma(E,E')$-$K$-convergent sequences. In particular, $T'$ is bounded with respect to these topologies.

A special case is obtained when $E$ is a normed $K$-space. Then $T'$ maps weak-$*$ bounded subsets of $D(T')$ to norm-bounded subsets of $E'$. In particular, $T'$ is norm-bounded.

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