---- > [!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 > [!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 > ```