| dbp:proof
|
- Suppose that is not dense in X.
By the Hahn–Banach theorem, there exists a non-zero that vanishes on .
For all x ∈ X,
:
Therefore, and is an eigenvalue of T*.
Conversely, suppose that is an eigenvalue of T*. Then there exists a non-zero such that , i.e.
:
If is dense in X, then φ must be the zero functional, a contradiction.
The claim is proved. (en)
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