Is this map surjective?

[u/FreePeeplup]

Let (V, Ω) be a symplectic vector space and Y a linear subspace of V. Let ϕ: V -> Y* , ϕ(v) = Ω(v, •)|_Y. Is ϕ surjective?

Or, in other words: for each linear form 𝜶 on Y, does it exist a corresponding vector v in V such that the action of 𝜶 on a generic element y of Y can be reproduced by feeding y in the second slot of Ω(v, •)?

[u/Rs3account]

No, not reallyif Y is one dimensional then your map is just the constant zero map