Integrals and Operators pp 93-122 | Cite as

# Convergence and Differentiation

## Abstract

The reader is probably familiar with the extensive and penetrating theory of finite-dimensional linear spaces and operators thereon. One cannot hope to obtain a similarly penetrating theory when the vector space is infinite-dimensional without additional structure in the space. A first step in civilizing a linear space for the purposes of analysis is the introduction of a convenient topology, i.e. one with respect to which relevant operations are continuous. A natural candidate for such a topology in the linear space **M** of all complex-valued locally measurable functions is that of sequential pointwise convergence. However, this topology may fail to have cogent structural properties. One is led rather to deal with a modification \(\tilde M\) of **M** which is obtained by identifying functions which agree except on local null sets; and which is given the topology it inherits naturally from that of pointwise sequential convergence in **M**. It will be seen in fact that \(\tilde M\) is metrizable in this topology, in the key case of a finite measure space; and that it becomes a topological linear space in the following sense.

## Keywords

Measurable Function Linear Space Measure Space Function Class Bounded Variation## Preview

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