definition of inner product on real or complex vectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of inner product on real or complex vectors space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( V\): \(\in \{\text{ the vectors spaces over } F\}\)
\(*\langle \bullet, \bullet\rangle\): \(: V \times V \to F\)
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Conditions:
\(\forall v_1, v_2, v_3 \in V, \forall r_1, r_2 \in F\)
(
1) \((0 \le \langle v_1, v_1 \rangle)\) \(\land\) \((0 = \langle v_1, v_1 \rangle \iff v_1 = 0)\)
2) \(\langle v_1, v_2 \rangle = \overline{\langle v_2, v_1 \rangle}\), where the over-line denotes the complex conjugate
3) \(\langle r_1 v_1 + r_2 v_2, v_3 \rangle = r_1 \langle v_1, v_3 \rangle + r_2 \langle v_2, v_3 \rangle\)
)
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2: Note
Inevitably, \(\langle v_3, r_1 v_1 + r_2 v_2 \rangle = \overline{\langle r_1 v_1 + r_2 v_2, v_3 \rangle} = \overline{r_1 \langle v_1, v_3 \rangle + r_2 \langle v_2, v_3 \rangle} = \overline{r_1} \overline{\langle v_1, v_3 \rangle} + \overline{r_2} \overline{\langle v_2, v_3 \rangle} = \overline{r_1} \langle v_3, v_1 \rangle + \overline{r_2} \langle v_3, v_2 \rangle\).
