Annihilator (mathematics)

There are two formulations of mathematics , which are denoted with the word annullator (or annihilator ).

Annulator in the context of forms

The cancellation space is a generalization of the orthogonal complement to vector spaces, in which the dual space cannot be identified with the space itself via a scalar product .

definition

Let be a vector space , the corresponding dual space and a subset of . Then is called ${\ displaystyle V}$ ${\ displaystyle V ^ {*}}$ ${\ displaystyle S}$ ${\ displaystyle V}$

{\ displaystyle S ^ {0} = \ lbrace f \ in V ^ {*} \ mid f (x) = 0 {\ text {for all}} x \ in S \ rbrace \ subseteq V ^ {*}}

the canceler of . ${\ displaystyle S}$

Features of the canceler

${\ displaystyle S ^ {0}}$ is a subspace of the dual space . That is why one speaks of the cancellation room . ${\ displaystyle V ^ {*}}$

${\ displaystyle S ^ {0} = \ langle S \ rangle ^ {0}}$ , where is the subspace generated by . ${\ displaystyle \ langle S \ rangle}$ ${\ displaystyle S}$

Is so is .

{\ displaystyle S_ {1} \ subseteq S_ {2}}

{\ displaystyle S_ {1} ^ {0} \ supseteq S_ {2} ^ {0}}

If is finite-dimensional and a subspace of , then we have . In this case and the dual space are canonically isomorphic and it applies , where and have been identified with one another. ${\ displaystyle V}$ ${\ displaystyle U}$ ${\ displaystyle V}$ ${\ displaystyle \ dim U ^ {0} = \ dim V- \ dim U}$ ${\ displaystyle V}$ ${\ displaystyle V ^ {**}}$ ${\ displaystyle \ left (U ^ {0} \ right) ^ {0} = U}$ ${\ displaystyle V}$ ${\ displaystyle V ^ {**}}$