The Bessel inequality lets us approximate norms in any inner product space or Hermitian product space.
Theorem. Let H be an inner product space or a Hermitian product space together with its product function ⟨.,.⟩. Let e1, e2, ... be any (finite or infinite) orthonormal sequence. Then for any x∈H,
∑i≥1 |⟨x,ei⟩|2 ≤
⟨x,x⟩=||x||2
When H is a Hilbert space, we can find in it complete orthonormal sequences; Parseval's theorem then lets us change inequality to equality.
Quantum mechanics take some joy in the Bessel inequality. It says that we will underestimate the energy of a system if we fail to take into account all possible states -- this makes sense. Similarly, probability theorists take great joy in noting that various independent random variables which are conditional expectations of the same random variable have a lower variance. In short -- Bessel's inequality makes a great deal of sense. It's nice to know it's true.
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