definition of 'normed vectors spaces - linear isometries' category
Topics
About: category
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of category.
- The reader knows a definition of 'normed vectors space' isometry.
- The reader knows a definition of linear map.
Target Context
- The reader will have a definition of 'normed vectors spaces - linear isometries' category.
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:
\(*C\): \(\in \{\text{ the categories }\}\), with \(Obj (C)\) and \(Mor (C)\) specified below
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Conditions:
\(Obj (C) = \{\text{ the normed vectors spaces }\} = \{O_j \vert j \in J\}\)
\(\land\)
\(Mor (C) = \{Mor (O_j, O_l) \vert O_j, O_l \in Obj (C)\}\), where \(Mor (O_j, O_l) = \{\text{ the linear isometries from } O_j \text{ into } O_l\}\)
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2: Note
Let us see that \(C\) is indeed a category.
Let \(O_1, O_2, O_3, O_4 \in Obj (C)\) and \(f_1 \in Mor (O_1, O_2), f_2 \in Mor (O_2, O_3), f_3 \in Mor (O_3, O_4)\) be any.
1) \(f_2 \circ f_1 \in Mor (O_1, O_3)\): \(f_2 \circ f_1\) is from \(O_1\) into \(O_3\); \(f_2 \circ f_1\) is linear; \(f_2 \circ f_1\) is isometric, because for each \(v \in O_1\), \(\Vert f_2 \circ f_1 (v) \Vert = \Vert f_1 (v) \Vert\), because \(f_2\) is isometric, \(= \Vert v \Vert\), because \(f_1\) is isometric.
2) \(\exists id_{O_j} \in Mor (O_j, O_j) (f_1 \circ id_{O_1} = f_1 \land id_{O_2} \circ f_1 = f_1)\): take the identity map, \(id_{O_j}: O_j \to O_j\), which is indeed linear isometric, and \(f_1 \circ id_{O_1} = f_1\) and \(id_{O_2} \circ f_1 = f_1\).
3) \(f_3 \circ (f_2 \circ f_1) = (f_3 \circ f_2) \circ f_1\): that holds as a property of any map.
We cannot have "'normed vectors spaces - complex-conjugate-linear isometries' category", because 1) would not hold: \(f_2 \circ f_1\) would not be complex-conjugate-linear, because \(f_2 \circ f_1 (r v) = f_2 (\overline{r} f_1 (v)) = \overline{\overline{r}} f_2 (v) = r f_2 (v)\), not complex-conjugate-linear but linear.
