Natural way to find an example of banach spaces is to look at closed subspaces of Banach spaces. Initially, It was really hard to find examples of closed subspaces of $L^2(R)$. Then, my professor gave me this example.

For $ f \in L^1(R) \cap L^2(R)$ such that $\hat f(0)=0.$ Then closure( in $L^2(R)$) of translational invariant linear subspace containing $f$ is the closed subspace of $L^2(R)$.

Then the question arises, what are all closed subspaces of $L^2(R)$ ?

Can we characterize all closed subspaces of $L^2(R)$ ?

I would like to know more examples of closed subspaces of $L^2(R)$, if you are familiar with them.

P.S. There is nothing special about $L^2(R)$, question is also applicable to $L^p(R)$. But $L^2(R)$ has nice properties ( Fourier transformation isometry).