Topic summary
Weak limit
Extracted from the Wikipedia article Weak topology.
Hilbert spaces
forms an orthonormal basis. In particular, the (strong) limit of as k → ∞ does not exist. On the other hand, being an orthonormal sequence in a Hilbert space, it must converge weakly to zero, by the Riesz representation theorem, Bessel's inequality, and the nth-term test for series. Alternatively, the Riemann–Lebesgue lemma can be used to show that the weak limit exists and is zero.
