If $\;\;V=\{ f:\mathbb{R}\rightarrow \mathbb{C} |\; f \text{ is continuous and has period }1\}$, $\;\; \langle f | g \rangle$ is defined as $ \langle f | g \rangle = \int_0^1 \overline{f(t)}g(t)dt$, $\forall f,g \in V\;\;$ and $\;\;H_a =\left\{g\in V: g\left(t+\frac{1}{2}\right)=g(t) \right\}\;\;\;$ (period $\frac{1}{2}$).
What can be said about $H_a^\perp$?
$H_a^\perp = \{f\in V: \langle f | g \rangle=0, \forall g\in H_a \} = \left\{f\in V: \langle f | g \rangle=0, \forall g: g\left(t+\frac{1}{2}\right)=g(t) \right\} $
Hints: $$0=\int_0^{1/2} f(x)g(x) \, dx+\int_{1/2}^{1} f(x)g(x) \, dx=\int_0^{1/2} f(x)g(x) \, dx+\int_{0}^{1/2} f(y-\frac 1 2)g(y-\frac 1 2) \, dy$$ $$=\int_0^{1/2} f(x)g(x) \, dx+\int_{0}^{1/2} f(y-\frac 1 2)g(y) \, dy$$ for all continuous functions $g$ on $[0,\frac 1 2]$ with $g(0)=g(\frac 1 2)$ iff $f(y-\frac 1 2)=-f(y)$. Thus the orthogonal complement consists of all continuous functions $f$ with period $1$ satisfying $f(x-\frac 1 2)=-f(x)$ for all $x$. Some details are missing but I think you can fill in the details.