Sobolev Space

For , an open subset of , and , the Sobolev space is defined by

(1)

where , , and the derivatives are taken in a weak sense.

When endowed with the norm

(2)

is a Banach space.

In the special case , is denoted by . This space is a Hilbert space for the inner product

(3)

Sobolev spaces play an important role in the theory of partial differential equations.