Is there a simple example of an infinite dimensional vector space such that $(W_1\cap W_2)^\circ \not= W_1^\circ +W_2^\circ$?
2026-04-11 23:23:51.1775949831
Counterexamples of Annihilator in Infinite Dimensional Setting
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This is not possible. If $E$ is a vector space and $W_1,W_2$ vector subspaces of $E$ then you have $$(W_1\cap W_2)^\circ = W_1^\circ +W_2^\circ.$$ This is true even if $E$ is of infinite dimension. However, Axiom of Choice is required in that case to work with basis.