Definition:Sobolev Space

From ProofWiki
Jump to navigation Jump to search

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 is defined and denoted as:



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

and equipped with the Sobolev norm.




Also see

  • Results about Sobolev spaces can be found here.


Source of Name

This entry was named for Sergei Lvovich Sobolev.


Sources