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Linear operator algebra
Definition
$X$…left $\mathcal R$-module |
$\langle \mathrm{Hom}(X,X),+,\cdot,*,\rangle \in L(X,X)$ |
$\langle \mathrm{Hom}(X,X),+,\cdot\rangle \in \mathcal L(X,X)$ |
$*:\mathrm{Hom}(X,X)\times \mathrm{Hom}(X,X)\to \mathrm{Hom}(X,X)$ |
$ v\in M $ |
$A,B \in \mathrm{Hom}(X,Y)$ |
$(A*B)v = A(B v) $ |
Discussion
Theorem: A linear operator $A:X\to X$ is bijective if it has an inverse in $L(X,X)$.