Let f ∊ L(X, Y) be a linear function between Hilbert spaces X and Y.
1. There exists a unique x* ∊ X such that fy(x) = xT x*.
2. Define f*: Y → X by f*(y) = x*. Then f * satisfies
f (x)T y = xT f*(y)
3. f* is a linear function, known as the adjoint off.