Editing Lp space (section)

===When {{math|(0 < ''p'' < 1)}}===

If <math>0 < p < 1,</math> then <math>L^p(\mu)</math> can be defined as above, that is:
<math display="block">N_p(f) = \int_S |f|^p\, d\mu < \infty.</math>
In this case, however, the <math>p</math>-norm <math>\|f\|_p = N_p(f)^{1/p}</math> does not satisfy the triangle inequality and defines only a [[quasi-norm]]. The inequality <math>(a + b)^p \leq a^p + b^p,</math> valid for <math>a, b \geq 0,</math> implies that
<math display="block">N_p(f + g) \leq N_p(f) + N_p(g)</math>
and so the function
<math display="block">d_p(f ,g) = N_p(f - g) = \|f - g\|_p^p</math>
is a metric on <math>L^p(\mu).</math> The resulting metric space is [[Complete metric space|complete]].{{sfn|Rudin|1991|p=37}}

In this setting <math>L^p</math> satisfies a ''reverse Minkowski inequality'', that is for <math>u, v \in L^p</math>
<math display="block">\Big\||u| + |v|\Big\|_p \geq \|u\|_p + \|v\|_p</math>

This result may be used to prove [[Clarkson's inequalities]], which are in turn used to establish the [[uniformly convex space|uniform convexity]] of the spaces <math>L^p</math> for <math>1 < p < \infty</math> {{harv|Adams|Fournier|2003}}.

The space <math>L^p</math> for <math>0 < p < 1</math> is an [[F-space]]: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is the prototypical example of an [[F-space]] that, for most reasonable measure spaces, is not [[Locally convex topological vector space|locally convex]]: in <math>\ell^p</math> or <math>L^p([0, 1]),</math> every open convex set containing the <math>0</math> function is unbounded for the <math>p</math>-quasi-norm; therefore, the <math>0</math> vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space <math>S</math> contains an infinite family of disjoint measurable sets of finite positive measure.

The only nonempty convex open set in <math>L^p([0, 1])</math> is the entire space. Consequently, there are no nonzero continuous linear functionals on <math>L^p([0, 1]);</math> the [[continuous dual space]] is the zero space. In the case of the [[counting measure]] on the natural numbers (i.e. <math>L^p(\mu) = \ell^p</math>), the bounded linear functionals on <math>\ell^p</math> are exactly those that are bounded on <math>\ell^1</math>, i.e., those given by sequences in <math>\ell^\infty.</math> Although <math>\ell^p</math> does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.

Having no linear functionals is highly undesirable for the purposes of doing analysis. In case of the Lebesgue measure on <math>\Reals^n,</math> rather than work with <math>L^p</math> for <math>0 < p < 1,</math> it is common to work with the [[Hardy space]] {{math|''H{{i sup|p}}''}} whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the [[Hahn–Banach theorem]] still fails in {{math|''H{{i sup|p}}''}} for <math>p < 1</math> {{harv|Duren|1970|loc=§7.5}}.