Just note that .
(from Kreyszig p.200 sec. 3.9 Hilbert Adjoint Operators)
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Let H_1 and H_2 be Hilbert Spaces and
T: H_1 \longrightarrow H_2 a bounded linear operator.
If M_1 \subset H_1 and M_2 \subset H_2 are such that
T(M_1) \subset M_2, show that
M^{\perp}_1 \supset T^{\ast}(M_2^{\perp}).