When my lecture notes discuss the adjoint of an operator in Banach spaces, it is defined like this. The adjoint of an operator(adsbygoogle = window.adsbygoogle || []).push({});

[tex]T:X\to Y[/tex]

is an operator

[tex]T^*:Y^*\to X^*[/tex]

so that for all [itex]f\in Y^*[/itex] and [itex]x\in X[/itex]

[tex](T^* f)(x) = f(T x)[/tex].

But we get into Hilbert spaces, it is said to be given by the equation

[tex](Tf|g) = (f|T^*g)[/tex]

The Hilbert space is also a Banach space, so these definitions seem to be contradicting.

In fact my lecture notes are unfortunately messy. I cannot tell for sure what precisely are the definitions, but this is what it says, approximately. Any major misunderstandings could be pointed out. I can conclude that I'm understanding something wrong, because I'm not understanding what the adjoint really is.

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# Adjoint of an operator definition

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