What is lp? — Mathematics

Definition 8.3 University Mathematics — Year 3 · Chapter 8 — Banach Spaces and the Fundamental Theorems

The classical sequence spaces (over $K$ , indexed by $\N$ ):

\ell^p = \Bigl\{x = (x_n) : \norm x_p = \Bigl(\sum_n \abs{x_n}^p\Bigr)^{1/p} < \infty\Bigr\}\quad (1 \leq p < \infty),

\ell^\infty = \{x : \norm x_\infty = \sup\abs{x_n} < \infty\},

and $c_0 = \{x : x_n \to 0\}$ with $\norm\cdot_\infty$ . That $\norm\cdot_p$ is a norm follows from the Minkowski inequality, proved in the discrete case exactly as in Chapter 12 (or by summing the finite-dimensional inequality of Year 2). All are Banach spaces, and $c_0$ is a closed subspace of $\ell^\infty$ (Exercise 8.3).

Unit balls of the p-norms in the plane, nested as p grows from 1 (diamond) through 2 (disc) and 4 (superellipse) to ∈fty (square). Convexity of every ball is the Minkowski inequality; the corners at p = 1 and p = ∈fty are where strict convexity, uniqueness of best approximations, and the equality cases of all degenerate at once. — Unit balls of the $p$ -norms in the plane, nested as $p$ grows from $1$ (diamond) through $2$ (disc) and $4$ (superellipse) to $\infty$ (square). Convexity of every ball is the Minkowski inequality; the corners at $p = 1$ and $p = \infty$ are where strict convexity, uniqueness of best approximations, and the equality cases of Exercise 12.12 all degenerate at once.

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