Adjoint Operators and Dual Spaces

Recall that A∗ : D(A∗) ⊂ X → X is defined in Definition 3.2 and we required that A be densely defined in X. We argue that this defines A∗ uniquely if D(A) is dense in X. To show this, suppose there exist w1 and w2 such that

〈Ax, y〉 = 〈x,w1〉 for all x ∈ D(A)

and 〈Ax, y〉 = 〈x,w2〉 for all x ∈ D(A).