Fredholm kernel

In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are one of the objects of study in Fredholm theory. Fredholm kernels are named in honour of Erik Ivar Fredholm. Much of the abstract theory of Fredholm kernels was developed by Alexander Grothendieck and published in 1955.

Let B be an arbitrary Banach space, and let B^* be its dual, that is, the space of bounded linear functionals on B. The tensor product ${\displaystyle B^{*}\otimes B}$ has a completion under the norm

where the infimum is taken over all finite representations

The completion, under this norm, is often denoted as

and is called the projective topological tensor product. The elements of this space are called Fredholm kernels.

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