Finite-rank truncation of a compact Hermitian operator

T x = \sum_{n} λ_{n} ⟨ x, e_{n} ⟩ e_{n}, λ_{n} \to 0,

then truncating the sum after

N

terms gives finite-rank Hermitian

T_{N}

with

∥ T - T_{N} ∥ = sup_{n > N} ∣ λ_{n} ∣ \to 0.

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