Understanding the Adjoint of a Linear Transformation on an Inner Product Space

[pivoxa15]

Definition: Let f:V->V be a linear transformation on an inner product space V. The adjoint f* of f is a linear transformation f*:V->V satisfying
<f(v),w>=<v,f*(w)> for all v,w in V.


My question is would <f*(v),w>=<v,f(w)> be equivalent to the above formula in the definition? If so why?

where <,> denote inner products.

 

[HallsofIvy]

Yes, it would be equivalent to your restricted definition. However, a more common, more general definition of "adjoint" is
If f is a linear transformation from one inner product space, U, to another inner product space, V, then the adjoint of f, f*, is the function from V to U such that
<f(u),v>_V= <u, f*(v)>_U. The subscripts indicate that < , >_V is the inner product in V, < , >_U is the inner product in U. Of course, since f is from U to V, in order for f(u) to be defined, u must be in U and then f(u) is in V. Conversely, in order for f*(v) to be defined v must be in V and then f*(v) is in U.
In that case, you could not just reverse the inner product. It is still true, however, that "adjoint" is a "dual" concept; the adjoint of f* is f itself.

(Oh, and the very important special case of "self-adjoint" only applies in the case of f:U-> U.)