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Proposition: Properties of a Complex Scalar Product

Let \(V\) be a vector space over the field of complex numbers \(\mathbb C\). It follows immediately from the definition of scalar product \(\langle \cdot,\cdot\rangle\) on \(V\) that it is function. \[V\times V\longrightarrow \mathbb {C} ,\,(v,w)\longmapsto \left\langle v,w\right\rangle \,,\]

with the following properties:

1. \(\left\langle \lambda _{1}x_{1}+\lambda _{2}x_{2},y\right\rangle =\lambda _{1}\left\langle x_{1},y\right\rangle +\lambda _{2}\left\langle x_{2},y\right\rangle \,\) for all \(\lambda_{1},\lambda _{2}\in \mathbb {C} \), \(x_{1},x_{2},y\in V\),
2. \(\left\langle x, \lambda _{1}y_{1}+\lambda _{2}y_{2}\right\rangle = \lambda_{1}^*\left\langle x, y_{1}\right\rangle + \lambda_{2}^*\left\langle x, y_{2}\right\rangle \,\) for all \(\lambda _{1},\lambda _{2}\in \mathbb {C} \), \(x, y_{1},y_{2}\in V\), where \(\lambda_i^*\) is the complex conjugate of \(\lambda_i\)
3. \(\left\langle v,w\right\rangle =\left\langle w,v\right\rangle^* \,\) for all \(v,w\in V\).
4. \(\left\langle v,v\right\rangle \geq 0\) for all \(v\in V\) and \(\left\langle v,v\right\rangle =0\) if and only if \(v=0\).

If only the first three properties are fulfilled, then we call \(\langle \cdot,\cdot\rangle\) a Hermitian form.

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References

Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück