Partial pointwise-boundedness and an argument of Pták (Q1182546)

In 1968 Pták proved the following theorem: Let \(\{T_ a\): \(a\in A\}\) be a pointwise-bounded family of linear maps of a Banach space \(X\) into a normed linear space \(Y\) and each \(T_ a\) has a closed null space \(N_ a\). Then the family \(\{T_ a\): \(a\in A\}\) is equicontinuous on some finite intersection of the null spaces. The paper uses the Pták techniques to extend the uniform boundedness principle by weakening the hypothesis of pointwise-boundedness and by reducing the continuity requirement on the maps involved. Extensions of the Arzelà-Ascoli theorem are also proved as well as some interpretations of the result which may enable them to be extended to automatic continuity paradigma, which might be useful in functional analysis or systems theory.