An [[Inner Product]] space is a type of [[Vector Space]] such that an [[Inner Product]] is defined. $\huge \begin{align} \mathbf{V} &= \ba{\underbrace{V}_{\text{\small Set of vectors}},\underbrace{K}_{\text{\small Some Field}}, +, \cdot, \braket{\cdot, \cdot} } \\ \end{align} $ >[!example] Euclidean >The simplest [[Inner Product Space]] is the common [[Euclidean Vector Space]] composing of [[Vector|vectors]] in $\R^{n}$ and the [[Field]] $\R$, in this context the [[Inner Product]] is typically defined as the [[Dot Product]]. > >$\huge \braket{\vec u,\vec v} = \vec u \cdot \vec v = \sum_{i=1}^{n} u_{i}v_{i} $