Let $V_0\leq V$ be a subspace of the vector space $V$. If $annih(V_0)\leq ann(V_0)\leq V^*$ is its annihilator, then the dual of $annih(V_0)$ ann(V_0)$is isomorphic with the quotient space$V/V_0$:$annih(V_0) ann(V_0) \cong \frac{V}{V_0}$This link may be useful: http://en.wikipedia.org/wiki/Dual_space#Quotient_spaces_and_annihilators Update: Oops, after I added my answer, the page refreshed and I saw that you edited your comment. It seems that my answer does not apply to your question... Anyway, I see no reason to give it a different name than annihilator, even if it annihilates a subspace of$V^*$. It has the same definition. Update 2 It may depend on the context: In functional analysis some use the notion of pre-annihilator because it is important to distinguish$V^{**}$from$V$. Example: Dales, Aiena - Introduction to Banach Algebras, Operators and Harmonic Analysis. When$V^{**}\cong V$some don't necessarily make the distinction. Example: Marsden and Ratiu - The Breadth of Symplectic and Poisson Geometry But it may be a good practice to specify in what space a subspace is the annihilator of a given subspace: Example: In Automorphic Forms on GL(2), Jacquet and Langlands said: "If$\tilde V_2$is the annihilator of$V_1$in$\tilde V$then$V_1$is the annihilator of$\tilde V_2$in$V$. For non-commutative rings, ideals, semigroups, one specifies whether the annihilator is left or right. Examples: G. Gratzer - Universal Algebra Steven G. Krantz (Ed.) - Dictionary of Algebra, Arithmetic, and Trigonometry 2 The answer was updated after the question was updated. Let$V_0\leq V$be a subspace of the vector space$V$. If$annih(V_0)\leq V^*$is its annihilator, then the dual of$annih(V_0)$is isomorphic with the quotient space$V/V_0$:$annih(V_0) \cong \frac{V}{V_0}$This link may be useful: http://en.wikipedia.org/wiki/Dual_space#Quotient_spaces_and_annihilators Update: Oops, after I added my answer, the page refreshed and I saw that you edited your comment. It seems that my answer does not apply to your question... Anyway, I see no reason to give it a different name than annihilator, even if it annihilates a subspace of$V^*$. It has the same definition. 1 Let$V_0\leq V$be a subspace of the vector space$V$. If$annih(V_0)\leq V^*$is its annihilator, then the dual of$annih(V_0)$is isomorphic with the quotient space$V/V_0$:$annih(V_0) \cong \frac{V}{V_0}\$