 Claim  If "T" is a compact self-adjoint operator on a non-zero Hilbert space "H" and

then "m"("T") or −"m"("T") is an eigenvalue of "T".If , then "T" = 0 by the polarization identity, and this case is clear.Consider the function

Replacing "T" by −"T" if necessary, one may assume that the supremum of "f" on the closed unit ball "B" ⊂ "H" is equal to .