K-convergence

P. Antosik and J. Mikusinski have introduced a stronger form of sequential convergence (cf. also Sequential space), called $\mathcal K$-convergence, which has found applications in a number of areas of analysis. If $\{x_k\}$ is a sequence in a Hausdorff Abelian topological group $(G,\tau)$, then $\{x_k\}$ is $\tau$-$\mathcal K$-convergent if every subsequence of $\{x_k\}$ has a further subsequence $\{x_{n_k}\}$ such that the subseries $\sum_{k=1}^\infty x_{n_k}$ is $\tau$-convergent in $G$. Any $\tau$-$\mathcal K$-convergent sequence is obviously $\tau$-null ($\tau$ convergent to $0$), but the converse does not hold in general although it does hold in a complete metric linear space. A space in which null sequences are $\mathcal K$-convergent is called a $\mathcal K$-space; a complete metric linear space is a $\mathcal K$-space, but there are examples of normal $\mathcal K$-spaces that are not complete [a2].

One of the principal uses of the notion of $\mathcal K$-convergence is in formulating versions of some of the classical results of functional analysis without imposing completeness or barrelledness assumptions. A subset $B$ of a topological vector space $E$ is bounded if for every sequence $\{x_k\}\subset B$ and every null scalar sequence $\{t_k\}$, the sequence $\{t_kx_k\}$ is a null sequence in $E$. A stronger form of boundedness is obtained by replacing the condition that $\{t_kx_k\}$ be a null sequence by the stronger requirement that $\{t_kx_k\}$ is $\mathcal K$-convergent; sets satisfying this stronger condition are called $\mathcal K$-bounded. In general, bounded sets are not $\mathcal K$-bounded; spaces for which the bounded sets are $\mathcal K$-bounded are called $\mathcal A$-spaces. Thus, $\mathcal K$-spaces are $\mathcal A$-spaces but there are examples of $\mathcal A$-spaces that are not $\mathcal K$-spaces. Using the notion of $\mathcal K$-boundedness, a version of the uniform boundedness principle (cf. Uniform boundedness) can be formulated which requires no completeness or barrelledness assumptions on the domain space of the operators. If $E$ and $F$ are topological vector spaces and $\Gamma$ is a family of continuous linear operators from $E$ into $F$ which is pointwise bounded on $E$, then $\Gamma$ is uniformly bounded on $\mathcal K$-bounded subsets of $E$. If $E$ is a complete metric linear space, this statement generalizes the classical uniform boundedness principle for $F$-spaces since in this case $\Gamma$ is equicontinuous (cf. also Equicontinuity). Similar versions of the Banach–Steinhaus theorem and the Mazur–Orlicz theorem on the joint continuity of separately continuous bilinear operators are possible. See [a1] or [a3] for these and further results.

References

[a1]	P. Antosik, C. Swartz, "Matrix methods in analysis" , Lecture Notes Math. , 1113 , Springer (1985)
[a2]	C. Klis, "An example of a non-complete (K) space" Bull. Acad. Polon. Sci. , 26 (1978) pp. 415–420
[a3]	C. Swartz, "Infinite matrices and the gliding hump" , World Sci. (1996)