definition of finite-product inner product
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of product 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 finite-product 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: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{ \mathbb{R}, \mathbb{C} \}\), with the canonical field structure
\( \{V_1, ..., V_n\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\), with any inner products, \(\{\langle \bullet, \bullet \rangle_1, ..., \langle \bullet, \bullet \rangle_n\}\)
\( V_1 \times ... \times V_n\): \(= \text{ the product vectors space }\)
\(*\langle \bullet, \bullet \rangle\): \(: (V_1 \times ... \times V_n) \times (V_1 \times ... \times V_n) \to F\), \(\in \{\text{ the inner products for } V_1 \times ... \times V_n\}\)
//
Conditions:
\(\forall (v_1, ..., v_n), (v'_1, ..., v'_n) \in V_1 \times ... \times V_n (\langle (v_1, ..., v_n), (v'_1, ..., v'_n) \rangle = \langle v_1, v'_1 \rangle_1 + ... + \langle v_n, v'_n \rangle_n)\)
//
2: Note
Let us see that \(\langle \bullet, \bullet \rangle\) is indeed an inner product.
Let \(v_1 = (v_{1, 1}, ..., v_{1, n}), v_2 = (v_{2, 1}, ..., v_{2, n}), v_3 = (v_{3, 1}, ..., v_{3, n}) \in V_1 \times ... \times V_n\) be any; let \(r_1, r_2 \in F\) be any.
1) \((0 \le \langle v_1, v_1 \rangle)\) \(\land\) \((0 = \langle v_1, v_1 \rangle \iff v_1 = 0)\): \(0 \le \langle v_{1, 1}, v_{1, 1} \rangle_1 + ... + \langle v_{1, n}, v_{1, n} \rangle_n = \langle (v_{1, 1}, ..., v_{1, n}), (v_{1, 1}, ..., v_{1, n}) \rangle = \langle v_1, v_1 \rangle\); when \(0 = \langle v_1, v_1 \rangle\), each \(\langle v_{1, j}, v_{1, j} \rangle_j = 0\), so, each \(v_{1, j} = 0\), so, \(v_1 = (v_{1, 1}, ..., v_{1, n}) = 0\); when \(v_1 = (v_{1, 1}, ..., v_{1, n}) = 0\), each \(v_{1, j} = 0\), so, each \(\langle v_{1, j}, v_{1, j} \rangle_j = 0\), so, \(0 = \langle v_1, v_1 \rangle\).
2) \(\langle v_1, v_2 \rangle = \overline{\langle v_2, v_1 \rangle}\), where the over-line denotes the complex conjugate: \(\langle v_1, v_2 \rangle = \langle v_{1, 1}, v_{2, 1} \rangle_1 + ... + \langle v_{1, n}, v_{2, n} \rangle_n = \overline{\langle v_{2, 1}, v_{1, 1} \rangle_1} + ... + \overline{\langle v_{2, n}, v_{1, n} \rangle_n} = \overline{\langle v_{2, 1}, v_{1, 1} \rangle_1 + ... + \langle v_{2, n}, v_{1, n} \rangle_n} = \overline{\langle v_2, v_1 \rangle}\).
3) \(\langle r_1 v_1 + r_2 v_2, v_3 \rangle = r_1 \langle v_1, v_3 \rangle + r_2 \langle v_2, v_3 \rangle\): \(\langle r_1 v_1 + r_2 v_2, v_3 \rangle = \langle r_1 (v_{1, 1}, ..., v_{1, n}) + r_2 (v_{2, 1}, ..., v_{2, n}), (v_{3, 1}, ..., v_{3, n}) \rangle = \langle (r_1 v_{1, 1} + r_2 v_{2, 1}, ..., r_1 v_{1, n} + r_2 v_{2, n}), (v_{3, 1}, ..., v_{3, n}) \rangle = \langle r_1 v_{1, 1} + r_2 v_{2, 1}, v_{3, 1} \rangle_1 + ... + \langle r_1 v_{1, n} + r_2 v_{2, n}, v_{3, n} \rangle_n = r_1 \langle v_{1, 1}, v_{3, 1} \rangle_1 + r_2 \langle v_{2, 1}, v_{3, 1} \rangle_1 + ... + r_1 \langle v_{1, n}, v_{3, n} \rangle_n + r_2 \langle v_{2, n}, v_{3, n} \rangle_n = r_1 (\langle v_{1, 1}, v_{3, 1} \rangle_1 + ... + \langle v_{1, n}, v_{3, n} \rangle_n) + r_2 (\langle v_{2, 1}, v_{3, 1} \rangle_1 + ... + \langle v_{2, n}, v_{3, n} \rangle_n) = r_1 \langle v_1, v_3 \rangle + r_2 \langle v_2, v_3 \rangle\).
The product needs to be finite, because otherwise, the inner product may not be into \(F\).
