Periodic Sobolev space ($H^s({\mathbb{T}}^d)$)

The periodic Sobolev space $H^s({\mathbb{T}}^d)$ consists of periodic distributions whose Fourier coefficients obeyIn one dimension, $H^1(\mathbb{T})$ embeds continuously into $L^{\infty }(\mathbb{T})$ and is a Banach algebra.