# 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**, denoted by $\map {H^n} {a, b}$, is defined by

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

## Source of Name

This entry was named for Sergei Lvovich Sobolev.

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*: Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense