Definition:Sobolev Space
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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:
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- $\map {H^n} {a, b} := \set {f \in \map {L^2} {a, b} : f', \ldots, f^{\paren n} \in \map {L^2} {a, b} }$
and equipped with the Sobolev norm.
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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