In mathematics, weak convergence in a Hilbert space is the convergence of a sequence of points in the weak topology.

A sequence of points ${\displaystyle (x_{n})}$ in a Hilbert space H is said to converge weakly to a point x in H if

for all y in H. Here, ${\displaystyle \langle \cdot ,\cdot \rangle }$ is understood to be the inner product on the Hilbert space. The notation

is sometimes used to denote this kind of convergence.

The Hilbert space ${\displaystyle L^{2}[0,2\pi ]}$ is the space of the square-integrable functions on the interval ${\displaystyle [0,2\pi ]}$ equipped with the inner product defined by

(see L^p space). The sequence of functions ${\displaystyle f_{1},f_{2},\ldots }$ defined by

converges weakly to the zero function in ${\displaystyle L^{2}[0,2\pi ]}$, as the integral

tends to zero for any square-integrable function ${\displaystyle g}$ on ${\displaystyle [0,2\pi ]}$ when ${\displaystyle n}$ goes to infinity, which is by Riemann–Lebesgue lemma, i.e.