# Corollary: Properties of a Real Scalar Product

Let $$V$$ be a vector space over the field of real numbers $$\mathbb R$$. Then the scalar product $$\langle \cdot,\cdot\rangle$$ on $$V$$ fulfills the following properties:

1. linear in the first argument: $$\left\langle \lambda _{1}v+\lambda _{2}w,y\right\rangle =\lambda _{1}\left\langle v,y\right\rangle +\lambda _{2}\left\langle w,y\right\rangle \,$$ for all $$\lambda_{1},\lambda _{2}\in \mathbb {R}$$, $$v,w,y\in V$$,
2. linear in the second argument: $$\left\langle x, \lambda _{1}v+\lambda _{2}w\right\rangle =\lambda _{1}\left\langle x, v\right\rangle +\lambda _{2}\left\langle x, w\right\rangle \,$$ for all $$\lambda _{1},\lambda _{2}\in \mathbb {R}$$, $$x, v,w\in V$$,
3. symmetric: $$\left\langle v,w\right\rangle =\left\langle w,v\right\rangle \,$$ for all $$v,w\in V$$.
4. symmetric: $$\left\langle v,w\right\rangle =\left\langle w,v\right\rangle \,$$ for all $$v,w\in V$$.

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