View source for K-convergence

P. Antosik and J. Mikusinski have introduced a stronger form of sequential convergence (cf. also [[Sequential space|Sequential space]]), called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k1200102.png" />-convergence, which has found applications in a number of areas of analysis. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k1200103.png" /> is a sequence in a Hausdorff Abelian [[Topological group|topological group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k1200104.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k1200105.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k1200108.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k1200109.png" />-convergent if every subsequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001010.png" /> has a further subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001011.png" /> such that the subseries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001012.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001013.png" />-convergent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001014.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001015.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001016.png" />-convergent sequence is obviously <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001018.png" />-null (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001019.png" /> convergent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001020.png" />), 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001021.png" />-convergent is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001023.png" />-space; a complete metric linear space is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001024.png" />-space, but there are examples of normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001025.png" />-spaces that are not complete [[#References|[a2]]].

One of the principal uses of the notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001026.png" />-convergence is in formulating versions of some of the classical results of [[Functional analysis|functional analysis]] without imposing completeness or barrelledness assumptions. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001027.png" /> of a [[Topological vector space|topological vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001028.png" /> is bounded if for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001029.png" /> and every null scalar sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001030.png" />, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001031.png" /> is a null sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001032.png" />. A stronger form of boundedness is obtained by replacing the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001033.png" /> be a null sequence by the stronger requirement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001034.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001035.png" />-convergent; sets satisfying this stronger condition are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001037.png" />-bounded. In general, bounded sets are not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001038.png" />-bounded; spaces for which the bounded sets are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001039.png" />-bounded are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001041.png" />-spaces. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001042.png" />-spaces are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001043.png" />-spaces but there are examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001044.png" />-spaces that are not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001045.png" />-spaces. Using the notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001046.png" />-boundedness, a version of the uniform boundedness principle (cf. [[Uniform boundedness|Uniform boundedness]]) can be formulated which requires no completeness or barrelledness assumptions on the domain space of the operators. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001048.png" /> are topological vector spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001049.png" /> is a family of continuous linear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001050.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001051.png" /> which is pointwise bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001052.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001053.png" /> is uniformly bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001054.png" />-bounded subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001055.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001056.png" /> is a complete metric linear space, this statement generalizes the classical uniform boundedness principle for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001058.png" />-spaces since in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120010/k12001059.png" /> is equicontinuous (cf. also [[Equicontinuity|Equicontinuity]]). Similar versions of the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]] and the [[Mazur–Orlicz theorem|Mazur–Orlicz theorem]] on the joint continuity of separately continuous bilinear operators are possible. See [[#References|[a1]]] or [[#References|[a3]]] for these and further results.

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