description/proof of that inner product on real or complex vectors space induces norm
Topics
About: normed vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of inner product on real or complex vectors space.
- The reader knows a definition of norm on real or complex vectors space.
- The reader admits the Cauchy-Schwarz inequality for any real or complex inner-producted vectors space.
Target Context
- The reader will have a description and a proof of the proposition that any inner product on any real or complex vectors space induces a norm.
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
\(V\): \(\in \{\text{ the vectors spaces over } F\}\) with any inner product, \(\langle \bullet, \bullet \rangle\)
\(\Vert \bullet \Vert\): \(: V \to \mathbb{R}, v \mapsto \sqrt{\langle v, v \rangle}\)
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Statements:
\(\Vert \bullet \Vert \in \{\text{ the norms on } V\}\).
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2: Natural Language Description
For any field, \(F \in \{\mathbb{R}, \mathbb{C}\}\), and any vectors space, \(V\), over \(F\), with any inner product, \(\langle \bullet, \bullet \rangle\), \(\Vert \bullet \Vert: V \to \mathbb{R}, v \mapsto \sqrt{\langle v, v \rangle}\) is a norm.
3: Proof
For any \(v_1, v_2 \in V\) and any \(r \in F\), 1) \(0 \le \Vert v_1 \Vert = \sqrt{\langle v_1, v_1 \rangle}\) with the equality holding if and only if \(v_1 = 0\); 2) \(\Vert r v_1 \Vert = \sqrt{\langle r v_1, r v_1 \rangle} = \sqrt{r \langle v_1, r v_1 \rangle} = \sqrt{r \overline{\langle r v_1, v_1 \rangle}} = \sqrt{r \overline{r \langle v_1, v_1 \rangle}} = \vert r \vert \sqrt{\langle v_1, v_1 \rangle} = \vert r \vert \Vert v_1 \Vert\); 3) \(\Vert v_1 + v_2 \Vert = \sqrt{\langle v_1 + v_2, v_1 + v_2 \rangle} = \sqrt{\langle v_1, v_1 \rangle + \langle v_1, v_2 \rangle + \langle v_2, v_1 \rangle + \langle v_2, v_2 \rangle} = \sqrt{\langle v_1, v_1 \rangle + \langle v_1, v_2 \rangle + \overline{\langle v_2, v_1 \rangle} + \langle v_2, v_2 \rangle} \le \sqrt{\langle v_1, v_1 \rangle + 2 \vert \langle v_1, v_2 \rangle \vert + \langle v_2, v_2 \rangle} \le \sqrt{\langle v_1, v_1 \rangle + 2 \sqrt{\langle v_1, v_1 \rangle} \sqrt{\langle v_2, v_2 \rangle} + \langle v_2, v_2 \rangle}\), by the Cauchy-Schwarz inequality for any real or complex inner-producted vectors space, \(= \sqrt{(\sqrt{\langle v_1, v_1 \rangle} + \sqrt{\langle v_2, v_2 \rangle})^2} = \sqrt{\langle v_1, v_1 \rangle} + \sqrt{\langle v_2, v_2 \rangle} = \Vert v_1 \Vert + \Vert \Vert v_2 \Vert\).
