Norms of vectors

In an inner product space, the norm (AKA length) of a vector is defined as \({\left\Vert v\right\Vert\equiv\sqrt{\left\langle v,v\right\rangle }}\), leading to common relations such as the Cauchy-Schwarz inequality \({\left|\left\langle v,w\right\rangle \right|\leq\left\Vert v\right\Vert \left\Vert w\right\Vert}\) and the triangle inequality (see below), and letting us define the angle between vectors by \({\cos\theta\equiv\left\langle v,w\right\rangle /\left(\left\Vert v\right\Vert \left\Vert w\right\Vert \right)}\). An inner product then defines a special class of bases, the orthonormal bases \({\hat{e}_{\mu}}\) with \({\left\langle \hat{e}_{\mu},\hat{e}_{\nu}\right\rangle =\delta_{\mu\nu}}\) (\({\equiv1}\) if \({\mu=\nu}\), \({0}\) otherwise). If we then write \({v=v^{\mu}\hat{e}_{\mu}}\) and \({w=w^{\mu}\hat{e}_{\mu}}\), we have

\(\displaystyle \left\langle v,w\right\rangle =\sum_{\mu}v^{\mu*}w^{\mu}=v^{\dagger}w, \)

where in the first expression we take the complex conjugate of the components \({v^{\mu}}\), and the second is common in linear algebra, where we treat the vectors as column matrices of components, and the inner product is formed by matrix multiplication after taking the adjoint (hermitian conjugate) of the first matrix.

Alternatively, we can define a norm on a real vector space as a non-negative real function that only vanishes for the zero vector and satisfies \({\left\Vert av\right\Vert =\left|a\right|\left\Vert v\right\Vert}\) as well as the triangle inequality \({\left\Vert v+w\right\Vert \leq\left\Vert v\right\Vert +\left\Vert w\right\Vert}\). This makes the space a real normed vector space. Existence of a norm does not in general imply the existence of an inner product, but if a norm satisfies the parallelogram identity

\(\displaystyle \left\Vert v+w\right\Vert ^{2}+\left\Vert v-w\right\Vert {}^{2}=2\left(\left\Vert v\right\Vert {}^{2}+\left\Vert w\right\Vert {}^{2}\right), \)

an inner product can be obtained using the polarization identity

\(\displaystyle \left\langle v,w\right\rangle =\left(\left\Vert v+w\right\Vert {}^{2}-\left\Vert v-w\right\Vert {}^{2}\right)/4=\left(\left\Vert v\right\Vert {}^{2}+\left\Vert w\right\Vert {}^{2}-\left\Vert v-w\right\Vert {}^{2}\right)/2. \)

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