# 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.

Proofs: 1

Definitions: 1

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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 OsnabrÃ¼ck