definition of Euclidean inner product on Euclidean vectors space
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of Euclidean vectors space.
- The reader knows a definition of inner product on real or complex vectors space.
Target Context
- The reader will have a definition of Euclidean inner product on Euclidean 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:
\( \mathbb{R}^d\): \(= \text{ the Euclidean vectors space }\)
\(*\langle \bullet, \bullet \rangle\): \(= \in \{ \text{ the inner products on } \mathbb{R}^d\}\), \(:\mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}, (v_1, v_2) \mapsto \sum_{j = 1 \sim d} v_1^j v_2^j\)
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Conditions:
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2: Natural Language Description
For the Euclidean vectors space, \(\mathbb{R}^d\), the inner product, \(\langle \bullet, \bullet \rangle\), \(:\mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}, (v_1, v_2) \mapsto \sum_{j = 1 \sim d} v_1^j v_2^j\)
3: Note
It is indeed an inner product: 1) \(0 \le \langle v_1, v_1\rangle = \sum_{j = 1 \sim d} v_1^j v_1^j\) with the equality holding if and only if \(v_1 = 0\); 2) \(\langle v_1, v_2 \rangle = \sum_{j = 1 \sim d} v_1^j v_2^j = \sum_{j = 1 \sim d} v_2^j v_1^j = \langle v_2, v_1\rangle\); 3) \(\langle r_1 v_1 + r_2 v_2, v_3 \rangle = \sum_{j = 1 \sim d} (r_1 v_1^j + r_2 v_2^j) v_3^j = \sum_{j = 1 \sim d} r_1 v_1^j v_3^j + \sum_{j = 1 \sim d} r_2 v_2^j v_3^j = r_1 \langle v_1, v_3 \rangle + r_2 \langle v_2, v_3 \rangle\).
Although a Euclidean vectors space tends to be implicitly supposed to have the Euclidean inner product, it is not necessarily so.
