---- > [!definition] Definition. ([[Hilbert space]]) > A **Hilbert space** is an [[inner product space]] that is [[Banach space|Banach]], i.e. is [[complete]] with respect to the [[norm]] [[canonical norm induced by inner product|induced]] by the [[inner product]]. ^definition > [!definition] (Bra-Ket notation) > Assume $H$ is a [[complex numbers|complex]] Hilbert space, with convention that $\langle \_,\_ \rangle$ is conjugate-linear in its first argument and linear in its second argument. > > The [[Riesz Representation Theorem]] gives that the [[inner product]] $\langle \_,\_ \rangle$ on a Hilbert space $H$ is a [[perfect pairing]], such that any $v \in H$ identifies canonically [[dual vector space|with]] $v^{*}=\langle v, \_ \rangle \in H^{*}$. In physics, it is customary to denote $v \in H$ by $|v \rangle$ (a **ket**) and $\langle v, \_ \rangle \in H^{*}$ by $\langle v |$ (a **bra**). Various [[tensor of type (p,q)|contractions]] are then denoted by appropriate juxtaposition of bras and kets. For example, $\langle v | w \rangle:=v^{*}(w)=\langle v,w \rangle$, [[isomorphism for tensor product and homset for finite-dimensional spaces|while]] $| v \rangle \langle w |=v \otimes w^{*}=\langle w, \_ \rangle v$. > Bra-ket notation usually comes accompanied by other conventions too, e.g. a basis vector $e_{i}$ often gets written $|i\rangle$. So for instance $I_{N}=\sum_{n=1}^{N} | i \rangle \langle i |$ for $(e_{i})$ an [[orthonormal basis]]. Another slight abuse is that if $v$ is an eigenvector with eigenvalue $\lambda$, the notation $| \lambda \rangle$ will sometimes get used instead of $| v \rangle$. > [!basicexample] > - [[nontrivial Lp spaces are Hilbert iff p=2]] ([[TODO]]) > - Special cases: $\mathbb{F}^{n}$ and $\ell^{2}$ > - a [[closed set|closed]] [[linear subspace]] of a Hilbert space is again a Hilbert space [[complete|is again a Hilbert space]] > ^basic-example > [!basicnonexample] > $C([-1,1])$ carries the [[inner product]] $\langle f,g \rangle:=\int_{-1}^{1} fg \, d \lambda$, inducing $\|\cdot\|_{2}$. However, it is not [[closed set|closed]] as a [[subspace topology|subspace]] of $L^{2}([-1,1])$, and [[complete|therefore is not complete]]. > Indeed, recall the [[sequence]] $(f_{k})$ in $C([-1,1])$ from [[pointwise converge#^c6bab8|this example]] which [[pointwise converge|converges pointwise]] to $f=1_{\{ 0 \}}$. [[Dominated Convergence Theorem|DCT]] implies $(f_{k}) \to f$ in $L^{2}$, but $f \not \in C([-1, 1])$. ^nonexample ---- #### ---- #### 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 > ```