Let be Banach spaces, be given such that such that such that and . Show that is closed in .
My idea was to use the quotient Banach space of and use the First Isomorphism Theorem, but this isn't very direct. Is there a more direct method for proving this? Note that isn't necessarily injective.
I think that quotienting by ker(U) is probably essential, because it seems to be the only way to get round the difficulty described above. If you then consider the induced map , it is injective and still satisfies the condition , and so the argument suggested by Focus becomes valid.