A definition of real inner product
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 real inner product.
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: Definition
For any real vectors space, \(V\), any function, \(\langle \bullet, \bullet\rangle: V \times V \rightarrow \mathbb{R}\), such that for any vectors, \(v_1, v_2, v_3 \in V\) and any real numbers, \(r_1, r_2 \in \mathbb{R}\), 1) \(\langle v_1, v_1\rangle \ge 0\) with the equality holding if and only if \(v_1 = 0\) 2) \(\langle v_1, v_2\rangle = \langle v_2, v_1\rangle\) 3) \(\langle r_1 v_1 + r_2 v_2, v_3\rangle = r_1 \langle v_1, v_3\rangle + r_2 \langle v_2, v_3\rangle\)
