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.)

 

[tacman]

Yes, <f*(v),w>=<v,f(w)> is equivalent to the formula in the definition. This is because the inner product on an inner product space is linear in its first argument. In other words, for any fixed vector w, the map v -> <v,w> is a linear transformation. Therefore, we can apply this property to the definition of the adjoint and obtain <f*(v),w> = <v,f*(w)> for all v,w in V. Both formulas express the same property, just with the roles of v and w switched.