Let $M\subset L_2(0, 1)$ be the set of $L_2$-functions such that $$ \int_0^1u(x)~dx = a $$ $\forall u\in M$, and a constant $a\in\mathbb{C}$ (or $a\in\mathbb{R}$, although to ensure that all bases are covered, I've assumed that $a\in\mathbb{C})$. I've been tasked with determining the orthogonal complement to this set, and I was simply wondering whether or not the working is sufficient.
Note that $$ \int_0^1u(x)~dx = a\iff\int_0^1u(x)~dx - a = 0\iff\int_0^1u(x)~dx - \int_0^1a~dx = \int_0^1u(x) - a~dx = 0 $$ Therefore, $M\subset L_2(a, b)$ may be regarded as the set of all $L_2$ functions difference from $a\in\mathbb{C}$ integrates to zero. I wanted to use this fact to conclude that $M^\perp$ is simply the set of constant-valued functions, since multiplication of $u(x) - a$ by these constants wouldn't alter the value of the corresponding integral from $0$ to $1$, but of course that's not the same as said said constant not altering the value of the integral of $u(x)$, alone, from $0$ to $1$. That being said, I'm not entirely sure how to proceed from here. Is it even really possible to determine $M^\perp$? If so, I'd appreciate any help.
Thank you in advance.