Bochner integral

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In mathematics, the Bochner integral extends the definition of Lebesgue integral to functions which take values in a Banach space.

The theory of vector-valued functions is a chapter of mathematical analysis, concerned with the generalisation to functions taking values in a Banach space, or more general topological vector space, of the notions of infinite sum and integral. It includes as a particular case the idea of operator-valued function, basic in spectral theory, and this case provided much of the motivation around 1930. When the vectors lie in a space of finite dimension, everything typically can be done component-by-component.

Infinite sums of vectors in a Banach space B, which are a fortiori complete metric spaces, converge just when they are Cauchy sequences with respect to the norm of the space. This case, of functions from the natural numbers to B, presents no particular fresh difficulty. An integral of a vector-valued function with respect to a measure is often called a Bochner integral, for Salomon Bochner. Modern developments of the Lebesgue integral often include this case, which does not require major modification of the theory based on real-valued functions, assuming the development of integration doesn't overabuse ordering properties of the real line. For instance, the approach to the Lebesgue integral described in Wikipedia first defines integrals for positive functions and only then for real-valued functions by breaking them up into a difference of a positive and negative part. This technique does not generalize well: the need to integrate Banach-space valued functions right from the beginning forces better foundations for integration.