Definition:Sobolev Space

Definition
Let $\openint a b$ be an open real interval.

Let $\map {L^2} {a, b}$ be the Lebesgue space.

Let $f \in \map {L^2} {a, b}$.

Suppose that in the sense of distributions we have that:


 * $f, \ldots, f^{\paren n} \in \map {L^2} {a, b}$

where $n \in \N$.

Then the Sobolev space, denoted by $\map {H^n} {a, b}$, is defined by


 * $\map {H^n} {a, b} := \set {f \in \map {L^2} {a, b} : f', \ldots, f^{\paren n} \in \map {L^2} {a, b} }$

and equipped with the Sobolev norm.