Lp Spaces and the Inequalities They Rest On

We use real functions on intervals of positive length with Lebesgue measure and finite p >= 1. The Lp norm of f is the p-th root of the integral of |f|^p. A constant c on a set of measure m has norm |c| times m^(1/p), so an indicator's L1 norm is its set's measure. Minkowski says the norm of f + g is at most the norm of f plus the norm of g. This needs proof because a p-th root does not split over a sum.

Two functions differing only on a null set have a difference with norm 0. An indicator of a point or countable set is nonzero there but has integral 0. Definiteness means that only the zero vector has norm 0, so Lp identifies functions equal almost everywhere. Homogeneity means that multiplying any function by c multiplies its norm by |c|.

For real functions on an interval with Lebesgue measure, L2 has inner product integral of fg; zero inner product means orthogonal. Complete means every Cauchy sequence converges in the space. L2 is complete, so it is a Hilbert space. On positive-length intervals, the Lp norm for finite p >= 1 comes from an inner product only at p = 2, as the parallelogram test shows. For conjugates, solve 1/q = 1 minus 1/p: the conjugate of 4 is 4/3.

An inner-product norm obeys the parallelogram law: ||f+g||^2 + ||f-g||^2 = 2||f||^2 + 2||g||^2. For real Lp on [0, 1] with Lebesgue measure and finite p >= 1, use disjoint indicators of equal length m > 0. Each norm is m^(1/p); sum and difference have norm (2m)^(1/p). The law forces 2(2m)^(2/p) = 4m^(2/p), hence p = 2. At p = 2, integral of fg gives the norm's inner product. Zero inner product means orthogonal.