definition of complex-conjugate-linear map
Topics
About: module
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of complex-conjugate-linear map.
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:
\( \mathbb{C}\): with the canonical field structure
\( V_1\): \(\in \{\text{ the } \mathbb{C} \text{ vectors spaces }\}\)
\( V_2\): \(\in \{\text{ the } \mathbb{C} \text{ vectors spaces }\}\)
\(*f\): \(: V_1 \to V_2\)
//
Conditions:
\(\forall r_1, r_2 \in \mathbb{C}, \forall v_1, v_2 \in V_1 (f (r_1 v_1 + r_2 v_2) = \overline{r_1} f (v_1) + \overline{r_2} f (v_2))\)
//
2: Note
Often, it is called "anti-linear map", but we do not take that name, because 'complex-conjugate' does not seem "anti".
Often, it is called "conjugate-linear map", but we do not think we need to be so lazy: when we are on the consensus that the map is between some complex modules, "conjugate-linear map" may seem immediately understandable, but in general, the map is between some modules over a ring, and just "conjugate" does not necessarily evoke "complex-conjugate".
