- #1
julian
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Let F be a subspace of a real vector space V and let
G \subset V_C
i.e. a subspace of its complexification. Define the real subspace of G by
G_R := G \cap V.
There is a symplectic form w[u,v]. The annihilator subspace F^perp of V is defined by
F^perp = {v \in V : w[u,v] = 0 for all u \in F}
The annihilator subspace G^perp of V_C is defined by
G^perp = {v \in V_C : w[u,v] = 0 for all u \in G}.
Then the following results hold:
(F^perp)_C = (F_C)^perp,
It is easy to see that (F^perp)_C subset of (F_C)^perp. Could somone prove it the other way? I'm thinking F_C is larger than F so the annihilator condition is more restrictive?
thanks
G \subset V_C
i.e. a subspace of its complexification. Define the real subspace of G by
G_R := G \cap V.
There is a symplectic form w[u,v]. The annihilator subspace F^perp of V is defined by
F^perp = {v \in V : w[u,v] = 0 for all u \in F}
The annihilator subspace G^perp of V_C is defined by
G^perp = {v \in V_C : w[u,v] = 0 for all u \in G}.
Then the following results hold:
(F^perp)_C = (F_C)^perp,
It is easy to see that (F^perp)_C subset of (F_C)^perp. Could somone prove it the other way? I'm thinking F_C is larger than F so the annihilator condition is more restrictive?
thanks