Editing Lp space (section)

==''L<sup>p</sup>'' spaces and Lebesgue integrals==

An <math>L^p</math> space may be defined as a space of measurable functions for which the <math>p</math>-th power of the [[absolute value]] is [[Lebesgue integrable]], where functions which agree almost everywhere are identified. More generally, let <math>(S, \Sigma, \mu)</math> be a [[measure space]] and <math>1 \leq p \leq \infty.</math><ref group=note>The definitions of <math>\|\cdot\|_p,</math> <math>\mathcal{L}^p(S,\, \mu),</math> and <math>L^p(S,\, \mu)</math> can be extended to all <math>0 < p \leq \infty</math> (rather than just <math>1 \leq p \leq \infty</math>), but it is only when <math>1 \leq p \leq \infty</math> that <math>\|\cdot\|_p</math> is guaranteed to be a norm (although <math>\|\cdot\|_p</math> is a [[quasi-seminorm]] for all <math>0 < p \leq \infty,</math>).</ref> 
When <math>p \neq \infty</math>, consider the set <math>\mathcal{L}^p(S,\, \mu)</math> of all [[measurable function]]s <math>f</math> from <math>S</math> to <math>\Complex</math> or <math>\Reals</math> whose [[absolute value]] raised to the <math>p</math>-th power has a finite integral, or in symbols:{{sfn|Rudin|1987|p=65}}
<math display="block">\|f\|_p ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(\int_S |f|^p\;\mathrm{d}\mu\right)^{1/p} < \infty.</math>

To define the set for <math>p = \infty,</math> recall that two functions <math>f</math> and <math>g</math> defined on <math>S</math> are said to be {{em|equal [[almost everywhere]]}}, written {{em|<math>f = g</math> a.e.}}, if the set <math>\{s \in S : f(s) \neq g(s)\}</math> is measurable and has measure zero. 
Similarly, a measurable function <math>f</math> (and its [[absolute value]]) is {{em|bounded}} (or {{em|dominated}}) {{em|almost everywhere}} by a real number <math>C,</math> written {{em|<math>|f| \leq C</math> a.e.}}, if the (necessarily) measurable set <math>\{s \in S : |f(s)| > C\}</math> has measure zero. 
The space <math>\mathcal{L}^\infty(S,\mu)</math> is the set of all measurable functions <math>f</math> that are bounded almost everywhere (by some real <math>C</math>) and <math>\|f\|_\infty</math> is defined as the [[infimum]] of these bounds:
<math display="block">\|f\|_\infty ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \inf \{C \in \Reals_{\geq 0} : |f(s)| \leq C \text{ for almost every } s\}.</math> 
When <math>\mu(S) \neq 0</math> then this is the same as the [[essential supremum]] of the absolute value of <math>f</math>:{{refn|group=note|If <math>\mu(S) = 0</math> then <math>\operatorname{esssup}|f| = -\infty.</math>}}
<math display="block">\|f\|_\infty ~=~ \begin{cases}\operatorname{esssup}|f| & \text{if } \mu(S) > 0,\\ 0 & \text{if } \mu(S) = 0.\end{cases}</math> 

For example, if <math>f</math> is a measurable function that is equal to <math>0</math> almost everywhere<ref group=note name=Non0Value0Example>For example, if a non-empty measurable set <math>N \neq \varnothing</math> of measure <math>\mu(N) = 0</math> exists then its [[indicator function]] <math>\mathbf{1}_N</math> satisfies <math>\|\mathbf{1}_N\|_p = 0</math> although <math>\mathbf{1}_N \neq 0.</math></ref> then <math>\|f\|_p = 0</math> for every <math>p</math> and thus <math>f \in \mathcal{L}^p(S,\, \mu)</math> for all <math>p.</math> 

For every positive <math>p,</math> the value under <math>\|\,\cdot\,\|_p</math> of a measurable function <math>f</math> and its absolute value <math>|f| : S \to [0, \infty]</math> are always the same (that is, <math>\|f\|_p = \||f|\|_p</math> for all <math>p</math>) and so a measurable function belongs to <math>\mathcal{L}^p(S,\, \mu)</math> if and only if its absolute value does. Because of this, many formulas involving <math>p</math>-norms are stated only for non-negative real-valued functions. Consider for example the identity <math>\|f\|_p^r = \|f^r\|_{p/r},</math> which holds whenever <math>f \geq 0</math> is measurable, <math>r > 0</math> is real, and <math>0 < p \leq \infty</math> (here <math>\infty / r \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \infty</math> when <math>p = \infty</math>). The non-negativity requirement <math>f \geq 0</math> can be removed by substituting <math>|f|</math> in for <math>f,</math> which gives <math>\|\,|f|\,\|_p^r = \|\,|f|^r\,\|_{p/r}.</math> 
Note in particular that when <math>p = r</math> is finite then the formula <math>\|f\|_p^p = \||f|^p\|_1</math> relates the <math>p</math>-norm to the <math>1</math>-norm.

'''Seminormed space of <math>p</math>-th power integrable functions'''

Each set of functions <math>\mathcal{L}^p(S,\, \mu)</math> forms a [[vector space]] when addition and scalar multiplication are defined pointwise.<ref group=note>Explicitly, the vector space operations are defined by:
<math display="block">\begin{align}
(f+g)(x) &= f(x)+g(x), \\
(s f)(x) &= s f(x)
\end{align}</math>
for all <math>f, g \in \mathcal{L}^p(S,\, \mu)</math> and all scalars <math>s.</math> These operations make <math>\mathcal{L}^p(S,\, \mu)</math> into a vector space because if <math>s</math> is any scalar and <math>f, g \in \mathcal{L}^p(S,\, \mu)</math> then both <math>s f</math> and <math>f + g</math> also belong to <math>\mathcal{L}^p(S,\, \mu).</math></ref> 
That the sum of two <math>p</math>-th power integrable functions <math>f</math> and <math>g</math> is again <math>p</math>-th power integrable follows from <math display=inline>\|f + g\|_p^p \leq 2^{p-1} \left(\|f\|_p^p + \|g\|_p^p\right),</math><ref group=proof name=UpperBoundForNormOfSum>When <math>1 \leq p < \infty,</math> the inequality <math>\|f + g\|_p^p \leq 2^{p-1} \left(\|f\|_p^p + \|g\|_p^p\right)</math> can be deduced from the fact that the function <math>F : [0, \infty) \to \Reals</math> defined by <math>F(t) = t^p</math> is [[Convex function|convex]], which by definition means that <math>F(t x + (1 - t) y) \leq t F(x) + (1 - t) F(y)</math> for all <math>0 \leq t \leq 1</math> and all <math>x, y</math> in the domain of <math>F.</math> Substituting <math>|f|, |g|,</math> and <math>\tfrac{1}{2}</math> in for <math>x, y,</math> and <math>t</math> gives <math>\left(\tfrac{1}{2}|f| + \tfrac{1}{2}|g|\right)^p \leq \tfrac{1}{2} |f|^p + \tfrac{1}{2} |g|^p,</math> which proves that <math>(|f| + |g|)^p \leq 2^{p-1} (|f|^p + |g|^p).</math> The triangle inequality <math>|f + g| \leq |f| + |g|</math> now implies <math>|f + g|^p \leq 2^{p-1} (|f|^p + |g|^p).</math> The desired inequality follows by integrating both sides. <math>\blacksquare</math></ref> 
although it is also a consequence of ''[[Minkowski inequality|Minkowski's inequality]]'' 
<math display="block">\|f + g\|_p \leq \|f\|_p + \|g\|_p</math> 
which establishes that <math>\|\cdot\|_p</math> satisfies the [[triangle inequality]] for <math>1 \leq p \leq \infty</math> (the triangle inequality does not hold for <math>0 < p < 1</math>). 
That <math>\mathcal{L}^p(S,\, \mu)</math> is closed under scalar multiplication is due to <math>\|\cdot\|_p</math> being [[Absolute homogeneity|absolutely homogeneous]], which means that <math>\|s f\|_p = |s| \|f\|_p</math> for every scalar <math>s</math> and every function <math>f.</math> 

[[Absolute homogeneity]], the [[triangle inequality]], and non-negativity are the defining properties of a [[seminorm]]. 
Thus <math>\|\cdot\|_p</math> is a seminorm and the set <math>\mathcal{L}^p(S,\, \mu)</math> of <math>p</math>-th power integrable functions together with the function <math>\|\cdot\|_p</math> defines a [[seminormed vector space]]. In general, the [[seminorm]] <math>\|\cdot\|_p</math> is not a [[Norm (mathematics)|norm]] because there might exist measurable functions <math>f</math> that satisfy <math>\|f\|_p = 0</math> but are not {{em|identically}} equal to <math>0</math><ref group=note name=Non0Value0Example /> (<math>\|\cdot\|_p</math> is a norm if and only if no such <math>f</math> exists). 

'''Zero sets of <math>p</math>-seminorms'''

{{anchor|kernel}}
If <math>f</math> is measurable and equals <math>0</math> a.e. then <math>\|f\|_p = 0</math> for all positive <math>p \leq \infty.</math>
On the other hand, if <math>f</math> is a measurable function for which there exists some <math>0 < p \leq \infty</math> such that <math>\|f\|_p = 0</math> then <math>f = 0</math> almost everywhere. When <math>p</math> is finite then this follows from the <math>p = 1</math> case and the formula <math>\|f\|_p^p = \||f|^p\|_1</math> mentioned above. <!--(this formula itself follows from <math>\|f\|_p^r = \|f^r\|_{p/r},</math> which holds whenever <math>f \geq 0</math> is measurable, <math>r > 0</math> is real, and <math>0 < p \leq \infty</math> (where <math>\infty / r \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \infty</math> when <math>p = \infty</math>)). -->

Thus if <math>p \leq \infty</math> is positive and <math>f</math> is any measurable function, then <math>\|f\|_p = 0</math> if and only if <math>f = 0</math> [[almost everywhere]]. Since the right hand side (<math>f = 0</math> a.e.) does not mention <math>p,</math> it follows that all <math>\|\cdot\|_p</math> have the same [[zero set]] (it does not depend on <math>p</math>). So denote this common set by
<math display="block">\mathcal{N} \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \{f : f = 0 \ \mu\text{-almost everywhere} \} = \{f \in \mathcal{L}^p(S,\, \mu) : \|f\|_p = 0\} \qquad \forall \ p.</math>
This set is a vector subspace of <math>\mathcal{L}^p(S,\, \mu)</math> for every positive <math>p \leq \infty.</math> 

'''Quotient vector space'''

Like every [[seminorm]], the seminorm <math>\|\cdot\|_p</math> induces a [[Norm (mathematics)|norm]] (defined shortly) on the canonical [[Quotient space (linear algebra)|quotient vector space]] of <math>\mathcal{L}^p(S,\, \mu)</math> by its vector subspace 
<math display="inline">\mathcal{N} = \{f \in \mathcal{L}^p(S,\, \mu) : \|f\|_p = 0\}.</math>
This normed quotient space is called {{em|Lebesgue space}} and it is the subject of this article. We begin by defining the quotient vector space.

Given any <math>f \in \mathcal{L}^p(S,\, \mu),</math> the [[coset]] <math>f + \mathcal{N} \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \{f + h : h \in \mathcal{N}\}</math> consists of all measurable functions <math>g</math> that are equal to <math>f</math> [[almost everywhere]]. 
The set of all cosets, typically denoted by 
<math display="block">\mathcal{L}^p(S, \mu) / \mathcal{N} ~~\stackrel{\scriptscriptstyle\text{def}}{=}~~ \{f + \mathcal{N} : f \in \mathcal{L}^p(S, \mu)\},</math>
forms a vector space with origin <math>0 + \mathcal{N} = \mathcal{N}</math> when vector addition and scalar multiplication are defined by <math>(f + \mathcal{N}) + (g + \mathcal{N}) \;\stackrel{\scriptscriptstyle\text{def}}{=}\; (f + g) + \mathcal{N}</math> and <math>s (f + \mathcal{N}) \;\stackrel{\scriptscriptstyle\text{def}}{=}\; (s f) + \mathcal{N}.</math> 
This particular quotient vector space will be denoted by
<math display="block">L^p(S,\, \mu) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \mathcal{L}^p(S, \mu) / \mathcal{N}.</math>
Two cosets are equal <math>f + \mathcal{N} = g + \mathcal{N}</math> if and only if <math>g \in f + \mathcal{N}</math> (or equivalently, <math>f - g \in \mathcal{N}</math>), which happens if and only if <math>f = g</math> almost everywhere; if this is the case then <math>f</math> and <math>g</math> are identified in the quotient space. Hence, strictly speaking <math>L^p(S,\, \mu) </math> consists of [[equivalence class]]es of functions.{{sfn|Stein|Shakarchi|2012|p=2}}

'''The <math>p</math>-norm on the quotient vector space'''

Given any <math>f \in \mathcal{L}^p(S,\, \mu),</math> the value of the seminorm <math>\|\cdot\|_p</math> on the [[coset]] <math>f + \mathcal{N} = \{f + h : h \in \mathcal{N}\}</math> is constant and equal to <math>\|f\|_p;</math> denote this unique value by <math>\|f + \mathcal{N}\|_p,</math> so that:
<math display=block>\|f + \mathcal{N}\|_p \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \|f\|_p.</math> 
This assignment <math>f + \mathcal{N} \mapsto \|f + \mathcal{N}\|_p</math> defines a map, which will also be denoted by <math>\|\cdot\|_p,</math> on the [[Quotient space (linear algebra)|quotient vector space]] 
<math display="block">L^p(S, \mu) ~~\stackrel{\scriptscriptstyle\text{def}}{=}~~ \mathcal{L}^p(S, \mu) / \mathcal{N} ~=~ \{f + \mathcal{N} : f \in \mathcal{L}^p(S, \mu)\}.</math>
This map is a [[Norm (mathematics)|norm]] on <math>L^p(S, \mu)</math> called the {{em|{{visible anchor|p-norm|text=<math>p</math>-norm}}}}. 
The value <math>\|f + \mathcal{N}\|_p</math> of a coset <math>f + \mathcal{N}</math> is independent of the particular function <math>f</math> that was chosen to represent the coset, meaning that if <math>\mathcal{C} \in L^p(S, \mu)</math> is any coset then <math>\|\mathcal{C}\|_p = \|f\|_p</math> for every <math>f \in \mathcal{C}</math> (since <math>\mathcal{C} = f + \mathcal{N}</math> for every <math>f \in \mathcal{C}</math>). 

'''The Lebesgue <math>L^p</math> space'''

The [[normed vector space]] <math>\left(L^p(S, \mu), \|\cdot\|_p\right)</math> is called {{em|<math>L^p</math> space}} or the {{em|Lebesgue space}} of <math>p</math>-th power integrable functions and it is a [[Banach space]] for every <math>1 \leq p \leq \infty</math> (meaning that it is a [[complete metric space]], a result that is sometimes called the [[Riesz–Fischer theorem#Completeness of Lp, 0 < p ≤ ∞|Riesz–Fischer theorem]]). 
When the underlying measure space <math>S</math> is understood then <math>L^p(S, \mu)</math> is often abbreviated <math>L^p(\mu),</math> or even just <math>L^p.</math> 
Depending on the author, the subscript notation <math>L_p</math> might denote either <math>L^p(S, \mu)</math> or <math>L^{1/p}(S, \mu).</math> 

If the seminorm <math>\|\cdot\|_p</math> on <math>\mathcal{L}^p(S,\, \mu)</math> happens to be a norm (which happens if and only if <math>\mathcal{N} = \{0\}</math>) then the normed space <math>\left(\mathcal{L}^p(S,\, \mu), \|\cdot\|_p\right)</math> will be [[Linear map|linearly]] [[isometrically isomorphic]] to the normed quotient space <math>\left(L^p(S, \mu), \|\cdot\|_p\right)</math> via the canonical map <math>g \in \mathcal{L}^p(S,\, \mu) \mapsto \{g\}</math> (since <math>g + \mathcal{N} = \{g\}</math>); in other words, they will be, [[up to]] a [[linear isometry]], the same normed space and so they may both be called "<math>L^p</math> space".

The above definitions generalize to [[Bochner space]]s. 

In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of <math>\mathcal{N}</math> in <math>L^p.</math> For <math>L^\infty,</math> however, there is a [[Lifting theory|theory of lifts]] enabling such recovery.

===Special cases===
For <math>1 \leq p \leq \infty</math> the <math>\ell^p</math> spaces are a special case of <math>L^p</math> spaces; when <math>S</math> are the [[natural number]]s <math>\mathbb{N}</math> and <math>\mu</math> is the [[counting measure]]. More generally, if one considers any set <math>S</math> with the counting measure, the resulting <math>L^p</math> space is denoted <math>\ell^p(S).</math> For example, <math>\ell^p(\mathbb{Z})</math> is the space of all sequences indexed by the integers, and when defining the <math>p</math>-norm on such a space, one sums over all the integers. The space <math>\ell^p(n),</math> where <math>n</math> is the set with <math>n</math> elements, is <math>\Reals^n</math> with its <math>p</math>-norm as defined above. 

Similar to <math>\ell^2</math> spaces, <math>L^2</math> is the only [[Hilbert space]] among <math>L^p</math> spaces. In the complex case, the inner product on <math>L^2</math> is defined by
<math display="block">\langle f, g \rangle = \int_S f(x) \overline{g(x)} \, \mathrm{d}\mu(x).</math>
Functions in <math>L^2</math> are sometimes called '''[[square-integrable function]]s''', '''quadratically integrable functions''' or '''square-summable functions''', but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a [[Riemann integral]] {{harv|Titchmarsh|1976}}.

As any Hilbert space, every space <math>L^2</math> is linearly isometric to a suitable <math>\ell^2(I),</math> where the cardinality of the set <math>I</math> is the cardinality of an arbitrary basis for this particular <math>L^2.</math> 

If we use complex-valued functions, the space <math>L^\infty</math> is a [[commutative]] [[C*-algebra]] with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative [[von Neumann algebra]]. An element of <math>L^\infty</math> defines a [[bounded operator]] on any <math>L^p</math> space by [[multiplication operator|multiplication]].

===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}}.