SOLVED: 8_ Prove the Open Mapping Theorem for Frechet spaces 12.1. The Open Mapping Theorem: Let % be Banach spaces and let A 95 4 be continuous linear surjection: Then A( V) is open in % whenever V is open in 9 . SetB(a,r) :== {x € % : Ilx - all < r}; B(r) := B(0,r) , U(b,r) := {y € % : Ily - bll < r}; U(r) := U(0,r):2.4. Definition.Frechet space is a TVS 9 whose topology is defined by translation invariant metric d such that ( 9 , d is complete:
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