---- > [!definition] Definition. ([[positive definite bilinear form]]) > A [[bilinear map|bilinear form]] $B$ is said to be **positive definite** if it 'assigns to every nonzero vector positive length': $B(v,v)>0$ for all $v \neq 0$[^1]. In finite dimensions, equivalently the matrix of $B$ in any [[basis]] is [[positive definite matrix|positive definite]]. Replacing gt;$ with $\geq$ gives the notion of a **positive semidefinite** [[bilinear map|bilinear form]], characterized in finite [[dimension|dimensions]] by its [[matrix of a bilinear form|matrix]] (in any [[basis]]) being [[positive semidefinite matrix|positive semidefinite]]. > Note that any positive definite bilinear form is [[nondegenerate bilinear form|nondegenerate]]. The converse is false. > [!definition] Definition. (Pre-inner product) > > A [[symmetric multilinear map|symmetric]] positive definite [[bilinear map|bilinear form]] $\langle \_,\_ \rangle$ is an [[inner product]]. > > A [[symmetric multilinear map|symmetric]] positive semidefinite [[bilinear map|bilinear form]] $\langle \_, \_ \rangle_{\text{pre}}$ is called a **pre-inner product** or **semi-inner product**. Using [[Cauchy-Schwarz Inequality|Cauchy-Schwarz]], the [[orthogonal complement|kernel/radical]] of a pre-inner product $\langle \_,\_ \rangle_{\text{pre}}$ can be shown to equal[^1] $\operatorname{ker } \langle \_,\_ \rangle _{\text{pre}}=\{ v : \langle v,v \rangle_{\text{pre}}=0 \}.$ As with any bilinear form, [[quotient set|quotienting]] by the radical gives a [[nondegenerate bilinear form|nondegenerate]] [[bilinear map|bilinear form]] on the quotient space; in this case an [[inner product]] on the quotient. [^1]: Indeed, if $v \in V$ is such that $\langle v,v \rangle_{\text{pre}}=0$ for all $w \in V$, then $|\langle v, w \rangle_{\text{pre}}|^{2} \leq \|v\|_{\text{pre}}^{2} \|w\|_{\text{pre}}^{2}=0$ implying $\langle v,w \rangle_{\text{pre}}=0$ for all $w \in V$. The reverse inclusion is trivial. (well maybe we don't want to outsource there since we need an inner product on the quotient) "Being PSD is essentially equivalent to satisfying a [[Cauchy-Schwarz Inequality|Cauchy-Schwarz]]-type inequality" [^1]: In other places this is phrased as '$B(v,v) \geq 0$ with equality iff $v=0$. ---- #### $C^{\infty}(M) \otimes C^{\infty}(M) \to \mathbb{R}$ So i guess the thing i want would be (again still on a Riemannian manifold) modding out by degeneracy descends the map $C^{\infty}(M) \otimes C^{\infty}(M) \to \mathbb{R}$ to the earnest L2 inner product, ---- #### 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 > ``` . ---- #### ---- #### References > [!backlink] > {CODE_BLOCK_PLACEHOLDER} > [!frontlink] > {CODE_BLOCK_PLACEHOLDER}