Notes on L and Sobolev Spaces

Note that ‖f‖Lp(X) may take the value ∞. Definition 1.2. The space L(X) is the set L(X) = {f : X → R | ‖f‖Lp(X) <∞} . The space L(X) satisfies the following vector space properties: (1) For each α ∈ R, if f ∈ L(X) then αf ∈ L(X); (2) If f, g ∈ L(X), then |f + g| ≤ 2p−1(|f | + |g|) , so that f + g ∈ L(X). (3) The triangle inequality is valid if p ≥ 1. The most interesting cases are p = 1, 2,∞, while all of the L arise often in nonlinear estimates. Definition 1.3. The space l, called “little L”, will be useful when we introduce Sobolev spaces on the torus and the Fourier series. For 1 ≤ p <∞, we set