Definition:Sobolev Space

Definition

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

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

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

Suppose that in the sense of distributions we have that:

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

where $n \in \N$.

Then the Sobolev space is defined and denoted as:

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

and equipped with the Sobolev norm.

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In particular:

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

is an inner product space with:

${\innerprod u v}_{H^k} = \ds \sum_{i \mathop = 0}^k {\innerprod {u^{\paren i} } {v^{\paren i} } }_{L^2}$

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Also see

• Results about Sobolev spaces can be found here.

Source of Name

This entry was named for Sergei Lvovich Sobolev.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense