description/proof of parallelogram law on vectors space normed induced by inner product
Topics
About: vectors space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the parallelogram law on any vectors space normed induced by any 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
\(V\): \(\in \{\text{ the } F \text{ vectors spaces }\}\), with \(\Vert \bullet \Vert: V \to \mathbb{R}\) induced by \(\langle \bullet, \bullet \rangle: V \times V \to F\)
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Statements:
\(\forall v_1, v_2 \in V (\Vert v_1 + v_2 \Vert^2 + \Vert v_1 - v_2 \Vert^2 = 2 (\Vert v_1 \Vert^2 + \Vert v_2 \Vert^2))\)
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2: Proof
Whole Strategy: Step 1: expand \(\Vert v_1 + v_2 \Vert^2 = \langle v_1 + v_2, v_1 + v_2 \rangle\) and \(\Vert v_1 - v_2 \Vert^2 = \langle v_1 - v_2, v_1 - v_2 \rangle\), and see that the sum is \(2 (\langle v_1, v_1 \rangle + \langle v_1, v_1 \rangle) = 2 (\Vert v_1 \Vert^2 + \Vert v_2 \Vert^2)\).
Step 1:
\(\Vert v_1 + v_2 \Vert^2 = \langle v_1 + v_2, v_1 + v_2 \rangle = \langle v_1, v_1 \rangle + \langle v_2, v_2 \rangle + \langle v_1, v_2 \rangle + \langle v_2, v_1 \rangle\).
\(\Vert v_1 - v_2 \Vert^2 = \langle v_1 - v_2, v_1 - v_2 \rangle = \langle v_1, v_1 \rangle + \langle v_2, v_2 \rangle - \langle v_1, v_2 \rangle - \langle v_2, v_1 \rangle\).
So, \(\Vert v_1 + v_2 \Vert^2 + \Vert v_1 - v_2 \Vert^2 = 2 \langle v_1, v_1 \rangle + 2 \langle v_2, v_2 \rangle = 2 (\langle v_1, v_1 \rangle + \langle v_2, v_2 \rangle) = 2 (\Vert v_1 \Vert^2 + \Vert v_2 \Vert^2)\).
