definition of product norm on finite-'direct sum' of normed vectors spaces
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of direct sum of modules.
- The reader knows a definition of norm on real or complex vectors space.
Target Context
- The reader will have a definition of product norm on finite-'direct sum' of normed vectors spaces.
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
\( J\): \(\in \{\text{ the finite index sets }\}\)
\( \{V_j \vert j \in J\}\): \(V_j \in \{\text{ the } F \text{ vectors spaces }\}\) with any norm, \(\Vert \bullet \Vert_j\)
\( \oplus_{j \in J} V_j\): \(= \text{ the direct sum }\)
\(*\Vert \bullet \Vert\): \(: \oplus_{j \in J} V_j \to \mathbb{R}\), \(\in \{\text{ the norms over } \oplus_{j \in J} V_j\}\)
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Conditions:
\(\forall v = \times_{j \in J} v_j \in \oplus_{j \in J} V_j (\Vert v \Vert = \sqrt{\sum_{j \in J} {\Vert v_j \Vert_j}^2})\)
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2: Note
Let us see that \(\Vert \bullet \Vert\) is indeed a norm.
Let \(v, v' \in \oplus_{j \in J} V_j\) and \(r \in F\) be any.
1) (\(0 \le \Vert v \Vert\)) \(\land\) (\((0 = \Vert v \Vert) \iff (v = 0)\)); \(0 \le \sqrt{\sum_{j \in J} {\Vert v_j \Vert_j}^2} = \Vert v \Vert\); when \(v = 0\), for each \(j \in J\), \(v_j = 0\), so, \(\Vert v_j \Vert_j = 0\), so, \(\Vert v \Vert = \sqrt{\sum_{j \in J} {\Vert v_j \Vert_j}^2} = 0\); when \(\Vert v \Vert = 0\), for each \(j \in J\), \(\Vert v_j \Vert_j = 0\), so, \(v_j = 0\), and so, \(v = 0\).
2) \(\Vert r v \Vert = \vert r \vert \Vert v \Vert\): \(\Vert r v \Vert = \sqrt{\sum_{j \in J} {\Vert r v_j \Vert_j}^2} = \sqrt{\sum_{j \in J} (\vert r \vert \Vert v_j \Vert_j)^2} = \sqrt{\sum_{j \in J} \vert r \vert^2 {\Vert v_j \Vert_j}^2} = \vert r \vert \sqrt{\sum_{j \in J} {\Vert v_j \Vert_j}^2} = \vert r \vert \Vert v \Vert\).
3) \(\Vert v + v' \Vert \le \Vert v \Vert + \Vert v' \Vert\): \(\Vert v + v' \Vert = \sqrt{\sum_{j \in J} {\Vert v_j + v'_j \Vert_j}^2} \le \sqrt{\sum_{j \in J} (\Vert v_j \Vert_j + \Vert v'_j \Vert_j)^2} \le \sqrt{\sum_{j \in J} {\Vert v_j \Vert_j}^2} + \sqrt{\sum_{j \in J} {\Vert v'_j \Vert_j}^2}\), by the proposition that for any \(2\) \(n\)-sequences of real numbers, the square-root of the sum of the squares of the sums of the corresponding items of the sequences is equal to or smaller than the sum of the square-roots of the sums of the squares of the items of the sequences, \(= \Vert v \Vert + \Vert v' \Vert\).
