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linear_operator_algebra [2013/09/13 21:39] nikolaj |
linear_operator_algebra [2014/03/21 11:11] |
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| - | ===== Linear operator algebra ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $X$...left $\mathcal R$-module | | ||
| - | | @#FFBB00: $\langle \mathrm{Hom}(X,X),+,\cdot,*,\rangle \in L(X,X)$ | | ||
| - | |||
| - | | @#88DDEE: $\langle \mathrm{Hom}(X,X),+,\cdot\rangle \in \mathcal L(X,X)$ | | ||
| - | | @#88DDEE: $*:\mathrm{Hom}(X,X)\times \mathrm{Hom}(X,X)\to \mathrm{Hom}(X,X)$ | | ||
| - | |||
| - | | $ v\in M $ | | ||
| - | | $A,B \in \mathrm{Hom}(X,Y)$ | | ||
| - | |||
| - | | @#55EE55: $(A*B)v = A(B v) $ | | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | Theorem: A linear operator $A:X\to X$ is bijective if it has an inverse in $L(X,X)$. | ||
| - | ==== Parents ==== | ||
| - | === Subset of === | ||
| - | [[Unital associative algebra]] | ||
| - | === Refinement of === | ||
| - | [[Linear operator]] | ||
