---- > [!definition] Definition. ([[L2 inner product of differential forms]]) > Let $(M,g)$ be an [[orientable manifold|oriented]] [[Riemannian manifold|Riemannian]] [[smooth manifold|manifold]] of dimension $n$, determining a [[Riemannian volume form]] $\omega_{g} \in \Omega^{n}(M)$. > Let $\xi, \eta \in \Omega^{p}(M)$ be [[compact|compactly]] [[support|supported]] [[differential form|differential]]. We define their **$L^{2}$ inner product** [[inner product|as]] the [[integration of a compactly supported volume form on an oriented smooth manifold|integral]] $\langle \! \langle \xi, \eta \rangle \! \rangle_{L^{2}(\omega_{g})}:= \int _{M} \langle \xi, \eta \rangle_{g} \omega_{g},$ where the function $\langle \xi, \eta \rangle_{g}: M \to \mathbb{R}$ is defined as the pointwise [[inner product]] $\langle \xi, \eta \rangle_{g}(x)=\langle \xi(x), \eta(x) \rangle_{g}$ of [[algebra of alternating multilinear forms|alternating multilinear forms]]. > This makes $\Omega^{p}(M)$ into an [[inner product space]], though not a [[Hilbert space]] (e.g. look at $p=0$ and think about [[Lp density of test functions|smooth functions approximating an indicator]]). Its [[completion of a metric space|completion]] is denoted $L^{2}\Omega^{p}(M)$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```