Is $\{\operatorname{rect}(t-n)\}_{n\in\mathbb N}$ a complete orthonormal set in $L^2(\mathbb R)$? In $L^2(\mathbb R)$, with the inner product $$(f,g)=\int_\mathbb R f(t) \overline{g(t)}dt$$ the system of rect function...--prophetes.ai

Is $\{\operatorname{rect}(t-n)\}_{n\in\mathbb N}$ a complete orthonormal set in $L^2(\mathbb R)$? In $L^2(\mathbb R)$, with the inner product $$(f,g)=\int_\mathbb R f(t) \overline{g(t)}dt$$ the system of rect functions $\\{\operatorname{rect}(t-n)\\}_{n\in\mathbb N}$ is an orthonormal set. It is easily demonstrable. Is this system also complete in $L^2(\mathbb R)$? Thanks in advance.

No, it is not. Try to find a (nonzero) function, which is orthogonal to all your rectangle functions. For simplicity, you can find such $f$ which vanishes outside $(-1/2,1/2)$.