Sobolev Ball

A Sobolev ball is a subspace of a Sobolev space such that the functions have finite energy:

\[W^{k,2}([0, T], R) := \{f: [0, T] \rightarrow \mathbb{C} : \lvert \lvert f \lvert \lvert_{W^{k,2}} \leq R\]

Equivalently, one can think in terms of the basis coefficients:

\[\theta^{k, 2}([0, T], R) := \{ (\theta_z)_z : \sum_z \Bigg(1 + \big( \frac{2 \pi z}{T} \big)^{2k} \Bigg) \lvert \theta_z \lvert^2 \leq R\]

Geometrically, Sobolev balls are ellipsoids.