mean-square convergence

A sequence of partial sums converging to a function in the sense that the average squared difference between them shrinks to zero, even where they still disagree pointwise.

A sequence of functions $S_N$ converges in mean square to $f$ on $[a,b]$ when $\dfrac1{b-a}\displaystyle\int_a^b(f-S_N)^2\,dx\to0$ as $N\to\infty$ -- a weaker requirement than pointwise convergence at every point, since disagreement on a small enough set contributes a negligible amount to the integral.