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For a vector space $V$ and its subspace $W$, there is a vector space $W^{\perp}:= \{x\in V^{*} | \forall y\in W. x(y)=0\}$.

If $V$ is a finite dimensional inner product vector space, I think $W^\perp$ can be called orthogonal complement of $W$. However, what's the name of this in the general case?

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That's the annihilator of $W$. It applies to any subset of $V$, not just to subspaces.

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    $\begingroup$ Just to add a little more context: This idea also applies to general vector spaces, $V$, not just those with an inner product. If there is an inner product, and if $W$ is a subspace, then there's a nice isomorphism between the thing OP called $W^\perp$ and $W' = \{ u \in V \mid \forall y \in W, \langle u, v\rangle = 0 \}$, which is what most people mean by "$W^\perp$". The isomorphism $W' \to W^\perp$ is simply $u \mapsto (v \mapsto \langle u, v \rangle)$; of course, proving it's an isomorphism takes a little bit of work. $\endgroup$ – John Hughes Jul 23 '18 at 12:54

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