If E is a Banach space, E* is its dual space (i.e. elements of E* are linear functionals on E), then there is a standard notation for action of functional f \in E* on element x from E: instead of f(x). In this notation the definition of adjoint operator T* to operator T: X --> Y is the following: T* acts from Y* to X* and satisfies condition

= , or what is the same, (T*f)x = f(Tx).

Thanks for your valuable comments.

The comment of Mr. Kadets implies that in particular in reflexive Banach spaces, the adjoint acts on the same space, so it is rearly a similar situation to the Hilbert space case (at least in the real case). Manfred Wolff

I am sorry, but Professor's Wolff comment is misleading: in a reflexive space $X$ one has that $X = X^{**}$, i. e. the space is equal to its bidual, not to its dual! The situation, when $X$ can be identified with its dual space appears from time to time for spaces that are not Hilbert ones, but in the identification of $X$ and $X^*$ in such examples is less natural than in the Hilbert space setting. A typical example is the space $X = \ell_p \oplus \ell_q$, where $1/p + 1/q = 1, 1 < p

For general Banach spaces, the adjoint operator it is also defined, a topic that can be found in Functional Analysis books such as Kreyszig,s book.