Let $X=C[-1,1]$, and $L_k= \{ <t^{k+2i}, i=0,1,2,... > \} $. Define an inner product on $X$ with respect to $L_0 \perp L_1$.
Then confirm that $L_0 \perp L_1 $ on your inner product.
Can we just define $<f,g> = \int_{-1}^1 f(t)g(t) dt$?
This is definitely an inner product. And if $f \in L_0$ and $g \in L_1$ so $f=at^{2i}$ and $g=bt^{1+2i}$, plugging these into the formula and then integrating will give zero for all $a,b,$ constants.