1 - Classical Banach Spaces

Summary

To begin, recall that a Banach space is a complete normed linear space. That is, a Banach space is a normed vector space (X, ∥ · ∥) that is a complete metric space under the induced metric d(x, y) = ∥ x – y ∥. Unless otherwise specified, we'll assume that all vector spaces are over ℝ, although, from time to time, we will have occasion to consider vector spaces over ℂ.

What follows is a list of the classical Banach spaces. Roughly translated, this means the spaces known to Banach. Once we have these examples out in the open, we'll have plenty of time to fill in any unexplained terminology. For now, just let the words wash over you.

The Sequence Spacesl_pandc₀

Arguably the first infinite-dimensional Banach spaces to be studied were the sequence spaces l_p and c₀. To consolidate notation, we first define the vector space s of all real sequences x = (x_n) and then define various subspaces of s.

For each 1 ≤ p < ∞, we define

and take l_p to be the collection of those x ∈ s for which ∥ x ∥_p < ∞. The inequalities of Hölder and Minkowski show that l_p is a normed space; from there it's not hard to see that l_p is actually a Banach space.

The space l_p is defined in exactly the same way for 0 < p < 1 but, in this case, ∥ · ∥_p defines a complete quasi-norm. That is, the triangle inequality now holds with an extra constant; specifically, ∥ x + y ∥_p ≤ 2^1/p(∥ x∥_p + ∥ y ∥_y).