description/proof of that norm on complex vectors space can be regarded to be norm on canonical real vectors space
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 proposition that any norm on any complex vectors space can be regarded to be a norm on the canonical real 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:
\(V\): \(\in \{\text{ the complex vectors spaces }\}\)
\(\Vert \bullet \Vert\): \(: V \to \mathbb{R}\)
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Statements:
\(\Vert \bullet \Vert \in \{\text{ the norms for } V \text{ as the canonical real vectors space }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(\Vert \bullet \Vert\) satisfies the conditions to be a norm on \(V\) as the real vectors space.
Step 1:
Let \(v_1, v_2 \in V\) and \(r \in \mathbb{R}\) be any.
1) (\(0 \le \Vert v_1 \Vert\)) \(\land\) (\((0 = \Vert v_1 \Vert) \iff (v_1 = 0)\)): \(0 \le \Vert v_1 \Vert\) because it is so for \(V\) as the complex vectors space; \((0 = \Vert v_1 \Vert) \iff (v_1 = 0)\) because it is so for \(V\) as the complex vectors space.
2) \(\Vert r v_1 \Vert = \vert r \vert \Vert v_1 \Vert\): \(r \in \mathbb{C}\), so, \(\Vert r v_1 \Vert = \vert r \vert \Vert v_1 \Vert\).
3) \(\Vert v_1 + v_2 \Vert \le \Vert v_1 \Vert + \Vert v_2 \Vert\): \(\Vert v_1 + v_2 \Vert \le \Vert v_1 \Vert + \Vert v_2 \Vert\), because it is so for \(V\) as the complex vectors space.
