The inner product is a generalization of the [[Dot Product]] to any [[Vector Space]]. This operation returns a scalar (an element of the [[Field]] $F$ of the vector space). $\huge \braket{\cdot, \cdot}: V \times V \to F $ Here are the axioms that an [[Inner Product]] operation must follow: $\huge \begin{align} \braket{ f_{1}+ f_{2}, g } &= \braket{f, g} + \braket{f_{2}, g} \\ \braket{kf, g} &= k\braket{f, g} \\ \braket{f, g} &= \braket{g, f} \\ \braket{f, f} &\ge 0 \\ \braket{f,f} = 0 &\iff f = \vec 0\\ \end{align} $ In other words, they must satisfy these Axioms: - Must be [[Distributive Property|Distributive]] - Any [[Scalar]] can be pulled out of the product - Must be [[Commutative Property|Commutative]] - The [[Operation]] between two of the same object must be greater or equal to 0. >[!example] Example: Functional [[Vector Space]] Inner Product >Let $\iota$ be a [[Vector Space]] of real [[Function|functions]] with [[Domain]] of the range $[-\pi,\pi]$ and [[Codomain]] of the [[Real Numbers]]. > >$\huge \begin{align} >\iota^{2} \ba{-\pi,\pi} &= \begin{cases} >f: [-\pi,\pi] \to \R >\end{cases} \\ >\end{align}$ > >Because of the defining propery, we can note that $\int_{-\pi}^{\pi} f^{2}(x)\d x$ is [[Finite]]. > >$\huge \begin{align} >\braket{f,g} &= \int_{-\pi}^{\pi} f(x)g(x)\d x >\end{align}$ > > In this space, we can derive that the $\cos$ and $\sin$ functions are [[Orthogonal]] to one another. > >$\huge \begin{align} > \braket{\cos(x),\sin(x)}&=\int_{-\pi}^{\pi}\sin(x)\cos(x)\d x\\ >&= \int_{-\pi}^{\pi} \frac{\sin(2x)}{2}\d x\\ > &= 0\\ > \therefore \cos &\perp \sin \end{align}$