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