L2- Convergence vs. pointwise convergence

Question

I am reading about convergence of Fourier Series and the author uses a equality symbol "$=$" in the following equation instead of the usual "$\rightarrow$" to represent the convergence behavior of a error function $e(t)$.

My question is: In the following equation

$\int_{-\infty}^{\infty}|e(t)|^2dt=0$ (Note the "$=$" symbol)

Isn't necessary $e(t)=0$ for all $t$?

Thanks

Answer 1

As you can see, near $t = -2\pi, -\pi, 0, \pi, 2\pi$, the Fourier series of the square wave does not converge point-wisely, but the total error is still approaching $0$.

To gain some intuition, consider $e(t) = 1$ for $t = 0$, and $0$ for other $t$'s, and try to calculate its integral.