In mathematics, the  spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces.
They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group  they were first introduced by Frigyes Riesz .
spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces.
Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, finance, engineering, and other disciplines.
Applications
Statistics
In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of  metrics, and measures of central tendency can be characterized as solutions to variational problems.
In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the  norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its  norm (its Euclidean length).
Techniques which use an L1 penalty, like LASSO, encourage solutions where many parameters are zero.
Techniques which use an L2 penalty, like ridge regression, encourage solutions where most parameter values are small.
Elastic net regularization uses a penalty term that is a combination of the  norm and the  norm of the parameter vector.
Hausdorff–Young inequality
The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps  to  (or  to ) respectively, where  and .
This is a consequence of the Riesz–Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality.
By contrast, if , the Fourier transform does not map into .
Hilbert spaces
Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus.
The spaces  and  are both Hilbert spaces.
In fact, by choosing a Hilbert basis , i.e., a maximal orthonormal subset of  or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to  (same  as above), i.e., a Hilbert space of type .
The {{math|''p''}}-norm in finite dimensions
The length of a vector  in the -dimensional real vector space  is usually given by the Euclidean norm:
\left\| x \right\| _2 = \left( {x_1}^2 + {x_2}^2 + \dotsb + {x_n}^2 \right)^{1/2} .
The Euclidean distance between two points  and  is the length  of the straight line between the two points.
In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space.
An analogy to this is suggested by taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other.
The class of -norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.
Definition
For a real number , the -norm or -norm of  is defined by
\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dotsb + |x_n|^p \right) ^{1/p} .
The absolute value bars are unnecessary when  is a rational number and, in reduced form, has an even numerator.
The Euclidean norm from above falls into this class and is the -norm, and the -norm is the norm that corresponds to the rectilinear distance.
The -norm or maximum norm (or uniform norm) is the limit of the -norms for .
It turns out that this limit is equivalent to the following definition:
\left\| x \right\| _\infty = \max \left\{ |x_1|, |x_2|, \dotsc, |x_n| \right\}
See -infinity.
For all , the -norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or norm), which are that:
only the zero vector has zero length,
the length of the vector is positive homogeneous with respect to multiplication by a scalar (positive homogeneity), and
the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).
Abstractly speaking, this means that  together with the -norm is a Banach space.
This Banach space is the -space over .
Relations between {{math|''p''}}-norms
The grid distance or rectilinear distance (sometimes called the "Manhattan distance") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance).
Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:
\left\| x \right\| _2 \leq \left\| x \right\| _1 .
This fact generalizes to -norms in that the -norm  of any given vector  does not grow with :
for any vector  and real numbers  and .
(In fact this remains true for  and .)
For the opposite direction, the following relation between the -norm and the -norm is known:
\left\| x \right\|_1 \leq \sqrt{n} \left\| x \right\|_2 .
This inequality depends on the dimension  of the underlying vector space and follows directly from the Cauchy–Schwarz inequality.
In general, for vectors in  where :
\left\| x \right\| _p \leq \left\| x \right\| _r \leq n^{\frac{1}{r} - \frac{1}{p}} \left\| x \right\| _p .
This is a consequence of Hölder's inequality.
When {{math|0 < ''p'' < 1}}
In  for , the formula
\|x\| _p = \left( |x_1| ^p + |x_2| ^p + \cdots + |x_n|^p \right)^{1/p}
defines an absolutely homogeneous function for ; however, the resulting function does not define a norm, because it is not subadditive.
On the other hand, the formula
|x_1|^p + |x_2|^p + \dotsb + |x_n|^p
defines a subadditive function at the cost of losing absolute homogeneity.
It does define an F-norm, though, which is homogeneous of degree .
Hence, the function
d_p(x, y) = \sum_{i=1}^n |x_i - y_i|^p
defines a metric.
The metric space  is denoted by .
Although the -unit ball  around the origin in this metric is "concave", the topology defined on  by the metric  is the usual vector space topology of , hence  is a locally convex topological vector space.
Beyond this qualitative statement, a quantitative way to measure the lack of convexity of  is to denote by  the smallest constant  such that the multiple  of the -unit ball contains the convex hull of , equal to .
The fact that for fixed  we have
C_p(n) = n^{\frac{1}{p} - 1} \to \infty,\quad \text{as } n \to \infty
shows that the infinite-dimensional sequence space  defined below, is no longer locally convex.
When {{math|''p'' {{=}} 0}}
There is one  norm and another function called the  "norm" (with quotation marks).
The mathematical definition of the  norm was established by Banach's Theory of Linear Operations.
The space of sequences has a complete metric topology provided by the F-norm
(x_n) \mapsto \sum_n 2^{-n} \frac{|x_n|}{1 +|x_n|},
which is discussed by Stefan Rolewicz in Metric Linear Spaces.
The -normed space is studied in functional analysis, probability theory, and harmonic analysis.
Another function was called the  "norm" by David Donoho—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector .
Many authors abuse terminology by omitting the quotation marks.
Defining , the zero "norm" of  is equal to
|x_1|^0 + |x_2|^0 + \cdots + |x_n|^0 .
alt=An animated gif of p-norms 0.1 through 2 with a step of 0.05.
|thumb|An animated gif of p-norms 0.1 through 2 with a step of 0.05.
This is not a norm because it is not homogeneous.
For example, scaling the vector  by a positive constant does not change the "norm".
Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computing, information theory, and statistics–notably in compressed sensing in signal processing and computational harmonic analysis.
Despite not being a norm, the associated metric, known as Hamming distance, is a valid distance, since homogeneity is not required for distances.
The {{math|''p''}}-norm in infinite dimensions and {{math|''ℓ''{{i sup|''p''}}}} spaces
The sequence space {{math|''ℓ''{{i sup|''p''}}}}
The -norm can be extended to vectors that have an infinite number of components (sequences), which yields the space .
This contains as special cases:
, the space of sequences whose series is absolutely convergent,
, the space of square-summable sequences, which is a Hilbert space, and
, the space of bounded sequences.
The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate.
Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by:
\begin{align} & (x_1, x_2, \ldots, x_n, x_{n+1},\ldots)+(y_1, y_2, \ldots, y_n, y_{n+1},\ldots) \\ = {} & (x_1+y_1, x_2+y_2, \ldots, x_n+y_n, x_{n+1}+y_{n+1},\ldots), \\[6pt] & \lambda \cdot \left (x_1, x_2, \ldots, x_n, x_{n+1},\ldots \right) \\ = {} & (\lambda x_1, \lambda x_2, \ldots, \lambda x_n, \lambda x_{n+1},\ldots).
\end{align}
Define the -norm:
\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \cdots +|x_n|^p + |x_{n+1}| ^p + \cdots \right) ^{1/p}
Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, , will have an infinite -norm for .
The space  is then defined as the set of all infinite sequences of real (or complex) numbers such that the -norm is finite.
One can check that as  increases, the set  grows larger.
For example, the sequence
\left(1, \frac{1}{2}, \ldots, \frac{1}{n}, \frac{1}{n+1},\ldots\right)
is not in , but it is in  for , as the series
1^p + \frac{1}{2^p} + \cdots + \frac{1}{n^p} + \frac{1}{(n+1)^p}+\cdots,
diverges for  (the harmonic series), but is convergent for .
One also defines the -norm using the supremum:
\left\| x \right\| _\infty = \sup(|x_1|, |x_2|, \dotsc, |x_n|,|x_{n+1}|, \ldots)
and the corresponding space  of all bounded sequences.
It turns out that, page 16
\left\| x \right\| _\infty = \lim_{ p \to \infty } \left\| x \right\| _p
if the right-hand side is finite, or the left-hand side is infinite.
Thus, we will consider  spaces for .
The -norm thus defined on  is indeed a norm, and  together with this norm is a Banach space.
The fully general  space is obtained—as seen below—by considering vectors, not only with finitely or countably-infinitely many components, but with "arbitrarily many components"; in other words, functions.
An integral instead of a sum is used to define the -norm.
General ℓ<sup>''p''</sup>-space
In complete analogy to the preceding definition one can define the space \ell^p(I) over a general index set I (and 1\leq p < \infty) as
\ell^p(I)=\left\{(x_i)_{i\in I} \in \mathbb{K}^I : \sum_{i\in I} |x_i|^p < \infty \right\}\,,
where convergence on the right means that only countably many summands are nonzero (see also Unconditional convergence).
With the norm
\left\| x \right\| _p = \left(  \sum_{i\in I} |x_i|^p \right) ^{1/p}
the space \ell^p(I) becomes a Banach space.
In the case where I is finite with  n  elements, this construction yields  with the p-norm defined above.
If I is countably infinite, this is exactly the sequence space \ell^p defined above.
For uncountable sets I this is a non-separable Banach space which can be seen as the locally convex direct limit of \ell^p-sequence spaces.Rafael Dahmen, Gábor Lukács: Long colimits of topological groups I: Continuous maps and homeomorphisms.
in: Topology and its Applications Nr. 270, 2020.
Example 2.14
The index set I can be turned into a measure space by giving it the discrete σ-algebra and the counting measure.
Then the space \ell^p(I) is just a special case of the more general L^p-space (see below). '
'L<sup>p</sup>'' spaces and Lebesgue integrals
An  space may be defined as a space of measurable functions for which the p-th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified.
More generally, let  and  be a measure space.
Consider the set of all measurable functions from  to  or  whose absolute value raised to the -th power has a finite integral, or equivalently, that
\|f\|_p \equiv \left( \int_S |f|^p\;\mathrm{d}\mu \right)^{1/p}<\infty
The set of such functions forms a vector space, with the following natural operations:
\begin{align} (f+g)(x) &= f(x)+g(x), \\  (\lambda f)(x) &= \lambda f(x) \end{align}
for every scalar .
That the sum of two -th power integrable functions is again -th power integrable follows from the inequality
\|f + g\|_p^p \leq 2^{p-1} \left (\|f\|_p^p + \|g\|_p^p \right ).
(This comes from the convexity of t\mapsto t^p for p\ge1.)
In fact, more is true.
Minkowski's inequality says the triangle inequality holds for .
Thus the set of -th power integrable functions, together with the function , is a seminormed vector space, which is denoted by \mathcal{L}^p(S,\, \mu).
For , the space \mathcal{L}^{\infty}(S,\mu) is the space of measurable functions bounded almost everywhere, with the essential supremum of its absolute value as a norm:
\|f\|_\infty \equiv \inf \{ C\ge 0 : |f(x)| \le C \text{ for almost every } x \}.
As in the discrete case, if there exists  such that , then
\|f\|_\infty = \lim_{p\to\infty}\|f\|_p.
\mathcal{L}^p(S,\, \mu) can be made into a normed vector space in a standard way; one simply takes the quotient space with respect to the kernel of .
Since for any measurable function , we have that  if and only if  almost everywhere, the kernel of  does not depend upon ,
\mathcal{N} \equiv \{f : f = 0 \ \mu\text{-almost everywhere} \} = \ker(\|\cdot\|_p) \qquad\forall\ 1\leq p < \infty
In the quotient space, two functions  and  are identified if  almost everywhere.
The resulting normed vector space is, by definition,
L^p(S, \mu) \equiv \mathcal{L}^p(S, \mu) / \mathcal{N}
In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of \mathcal{N} in L^p.
For L^{\infty}, however, there is a theory of lifts enabling such recovery.
When the underlying measure space  is understood,  is often abbreviated , or just .
For  is a Banach space.
The fact that  is complete is often referred to as the Riesz-Fischer theorem, and can be proven using the convergence theorems for Lebesgue integrals.
The above definitions generalize to Bochner spaces.
Special cases
Similar to the  spaces,  is the only Hilbert space among  spaces.
In the complex case, the inner product on  is defined by
\langle f, g \rangle = \int_S f(x) \overline{g(x)} \, \mathrm{d}\mu(x)
The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics.
Functions in  are sometimes called quadratically integrable functions, square-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 .
If we use complex-valued functions, the space  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  defines a bounded operator on any  space by multiplication.
For  the  spaces are a special case of  spaces, when , and  is the counting measure on .
More generally, if one considers any set  with the counting measure, the resulting  space is denoted .
For example, the space  is the space of all sequences indexed by the integers, and when defining the -norm on such a space, one sums over all the integers.
The space , where  is the set with  elements, is  with its -norm as defined above.
As any Hilbert space, every space  is linearly isometric to a suitable , where the cardinality of the set  is the cardinality of an arbitrary Hilbertian basis for this particular .
Properties of ''L''<sup>''p''</sup> spaces
Dual spaces
The dual space (the Banach space of all continuous linear functionals) of  for  has a natural isomorphism with , where  is such that  (i.e. ).
This isomorphism associates  with the functional  defined by
f \mapsto \kappa_p(g)(f) = \int f g \, \mathrm{d}\mu\  for every f \in L^p(\mu)
The fact that  is well defined and continuous follows from Hölder's inequality.
is a linear mapping which is an isometry by the extremal case of Hölder's inequality.
It is also possible to show (for example with the Radon–Nikodym theorem, see, Theorem 6.16) that any  can be expressed this way: i.e., that  is onto.
Since  is onto and isometric, it is an isomorphism of Banach spaces.
With this (isometric) isomorphism in mind, it is usual to say simply that   is the dual Banach space of .
For , the space  is reflexive.
Let  be as above and let  be the corresponding linear isometry.
Consider the map from  to , obtained by composing  with the transpose (or adjoint) of the inverse of :
j_p : L^p(\mu) \mathrel{\overset{\kappa_q}{\longrightarrow}} L^q(\mu)^* \mathrel{\overset{\left(\kappa_p^{-1}\right)^*}{\longrightarrow}} L^p(\mu)^{**}
This map coincides with the canonical embedding  of  into its bidual.
Moreover, the map  is onto, as composition of two onto isometries, and this proves reflexivity.
If the measure  on  is sigma-finite, then the dual of  is isometrically isomorphic to  (more precisely, the map  corresponding to  is an isometry from  onto ).
The dual of  is subtler.
Elements of  can be identified with bounded signed finitely additive measures on  that are absolutely continuous with respect to .
See ba space for more details.
If we assume the axiom of choice, this space is much bigger than  except in some trivial cases.
However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo–Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of  is .
See Sections 14.77 and 27.44–47 Embeddings
Colloquially, if , then  contains functions that are more locally singular, while elements of  can be more spread out.
Consider the Lebesgue measure on the half line .
A continuous function in  might blow up near  but must decay sufficiently fast toward infinity.
On the other hand, continuous functions in  need not decay at all but no blow-up is allowed.
The precise technical result is the following.
Suppose that .
Then:
iff  does not contain sets of finite but arbitrarily large measure, and
iff  does not contain sets of non-zero but arbitrarily small measure.
Neither condition holds for the real line with the Lebesgue measure.
In both cases the embedding is continuous, in that the identity operator is a bounded linear map from  to  in the first case, and  to  in the second.
(This is a consequence of the closed graph theorem and properties of  spaces.)
Indeed, if the domain  has finite measure, one can make the following explicit calculation using Hölder's inequality
\ \|\mathbf{1}f^{p}\|_1 \le \|\mathbf{1}\|_{q/(q-p)} \|f^{p}\|_{q/p}
leading to
\ \|f\|_p \le \mu(S)^{1/p - 1/q} \|f\|_q .
The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity  is precisely
\|I\|_{q,p} = \mu(S)^{1/p - 1/q}
the case of equality being achieved exactly when  -almost-everywhere.
Dense subspaces
Throughout this section we assume that: .
Let  be a measure space.
An integrable simple function  on  is one of the form
f = \sum_{j=1}^n a_j \mathbf{1}_{A_j}
where  is scalar,  has finite measure and {\mathbf 1}_{A_j} is the indicator function of the set A_j, for .
By construction of the integral, the vector space of integrable simple functions is dense in .
More can be said when  is a normal topological space and  its Borel –algebra, i.e., the smallest –algebra of subsets of  containing the open sets.
Suppose  is an open set with .
It can be proved that for every Borel set  contained in , and for every , there exist a closed set  and an open set  such that
F \subset A \subset U \subset V \quad \text{and} \quad \mu(U) - \mu(F) = \mu(U \setminus F) < \varepsilon
It follows that there exists a continuous  Urysohn function  on  that is  on  and  on , with
\int_S |\mathbf{1}_A - \varphi| \, \mathrm{d}\mu < \varepsilon \ .
If  can be covered by an increasing sequence  of open sets that have finite measure, then the space of –integrable continuous functions is dense in .
More precisely, one can use bounded continuous functions that vanish outside one of the open sets .
This applies in particular when  and when  is the Lebesgue measure.
The space of continuous and compactly supported functions is dense in .
Similarly, the space of integrable step functions is dense in ; this space is the linear span of indicator functions of bounded intervals when , of bounded rectangles when  and more generally of products of bounded intervals.
Several properties of general functions in  are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions.
For example, it is proved this way that translations are continuous on , in the following sense:
\forall f \in L^p \left(\mathbf{R}^d\right):\quad \left\|\tau_t f - f \right\|_p \to 0,\quad \text{as } \mathbf{R}^d \ni t \to 0,
where
(\tau_t f)(x) = f(x - t).
{{math|''L<sup>p</sup>'' (0 < ''p'' < 1)}}
Let  be a measure space.
If , then  can be defined as above: it is the vector space of those measurable functions  such that
N_p(f) = \int_S |f|^p\, d\mu < \infty.
As before, we may introduce the -norm , but  does not satisfy the triangle inequality in this case, and defines only a quasi-norm.
The inequality , valid for  implies that
N_p(f + g)\le N_p(f) + N_p(g)
and so the function
d_p(f ,g) = N_p(f - g) = \|f - g\|_p^p
is a metric on .
The resulting metric space is complete; the verification is similar to the familiar case when .
In this setting  satisfies a reverse Minkowski inequality, that is for  in
\Big\||u| + |v|\Big\|_p \geq \|u\|_p + \|v\|_p
This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces  for  .
The space  for  is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous.
It is also locally bounded, much like the case .
It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in  or , every open convex set containing the  function is unbounded for the -quasi-norm; therefore, the  vector does not possess a fundamental system of convex neighborhoods.
Specifically, this is true if the measure space  contains an infinite family of disjoint measurable sets of finite positive measure.
The only nonempty convex open set in  is the entire space .
As a particular consequence, there are no nonzero linear functionals on : the dual space is the zero space.
In the case of the counting measure on the natural numbers (producing the sequence space ), the bounded linear functionals on  are exactly those that are bounded on , namely those given by sequences in .
Although  does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.
The situation of having no linear functionals is highly undesirable for the purposes of doing analysis.
In the case of the Lebesgue measure on , rather than work with  for , it is common to work with the Hardy space  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  for  .
{{math|''L''<sup>0</sup>}}, the space of measurable functions
The vector space of (equivalence classes of) measurable functions on  is denoted  .
By definition, it contains all the , and is equipped with the topology of convergence in measure.
When  is a probability measure (i.e., ), this mode of convergence is named convergence in probability.
The description is easier when  is finite.
If  is a finite measure on , the  function admits for the convergence in measure the following fundamental system of neighborhoods
V_\varepsilon = \Bigl\{ f : \mu \bigl(\{x : |f(x)| > \varepsilon \} \bigr) < \varepsilon \Bigr\}, \qquad \varepsilon > 0
The topology can be defined by any metric  of the form
d(f, g) = \int_S \varphi \bigl(|f(x) - g(x)|\bigr)\, \mathrm{d}\mu(x)
where  is bounded continuous concave and non-decreasing on , with  and  when  (for example, .
Such a metric is called Lévy-metric for .
Under this metric the space  is complete (it is again an F-space).
The space  is in general not locally bounded, and not locally convex.
For the infinite Lebesgue measure  on , the definition of the fundamental system of neighborhoods could be modified as follows
W_\varepsilon = \left\{ f : \lambda \left(\left\{ x : |f(x)| > \varepsilon \text{ and } |x| < \frac{1}{\varepsilon}\right\} \right) < \varepsilon \right\}
The resulting space  coincides as topological vector space with , for any positive –integrable density .
Generalizations and extensions
Weak {{math|''L<sup>p</sup>''}}
Let  be a measure space, and  a measurable function with real or complex values on .
The distribution function of  is defined for  by
\lambda_f(t) = \mu\left\{x\in S: |f(x)| > t\right\}
If  is in  for some  with , then by Markov's inequality,
\lambda_f(t)\le \frac{\|f\|_p^p}{t^p}
A function  is said to be in the space weak , or , if there is a constant  such that, for all ,
\lambda_f(t) \le \frac{C^p}{t^p}
The best constant  for this inequality is the -norm of , and is denoted by
\|f\|_{p,w} = \sup_{t > 0} ~ t \lambda_f^{1/p}(t) .
The weak  coincide with the Lorentz spaces , so this notation is also used to denote them.
The -norm is not a true norm, since the triangle inequality fails to hold.
Nevertheless, for  in ,
\|f\|_{p,w}\le \|f\|_p
and in particular .
In fact, one has
\| f \|^p_{L^p} = \int |f(x)|^p d\mu(x) \geq \int_{\{|f(x)| > t \}} t^p + \int_{\{|f(x)| \leq t \}} |f|^p \geq t^p \mu(\{|f| > t \}),
and raising to power  and taking the supremum in  one has
\|f\|_{L^p} \geq \sup_{t > 0} t \; \mu(\{|f| > t \})^{1/p} = \|f\|_{L^{p,w}} .
Under the convention that two functions are equal if they are equal  almost everywhere, then the spaces  are complete .
For any  the expression
||| f |||_{L^{p,\infty}}=\sup_{0<\mu(E)<\infty} \mu(E)^{-1/r + 1/p} \left(\int_E |f|^r\,d\mu\right)^{1/r}
is comparable to the -norm.
Further in the case , this expression defines a norm if .
Hence for  the weak  spaces are Banach spaces .
A major result that uses the -spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.
Weighted {{math|''L<sup>p</sup>''}} spaces
As before, consider a measure space .
Let  be a measurable function.
The -weighted  space is defined as , where  means the measure  defined by
\nu (A) \equiv \int_A w(x) \, \mathrm{d} \mu (x), \qquad A \in \Sigma,
or, in terms of the Radon–Nikodym derivative,  the norm for  is explicitly
\| u \|_{L^p (S, w \, \mathrm{d} \mu)} \equiv \left( \int_S w(x) |u(x)|^p \, \mathrm{d} \mu (x) \right)^{1/p}
As -spaces, the weighted spaces have nothing special, since  is equal to .
But they are the natural framework for several results in harmonic analysis ; they appear for example in the Muckenhoupt theorem: for , the classical Hilbert transform is defined on  where  denotes the unit circle and  the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on .
Muckenhoupt's theorem describes weights  such that the Hilbert transform remains bounded on  and the maximal operator on .
{{math|''L<sup>p</sup>''}} spaces on manifolds
One may also define spaces  on a manifold, called the intrinsic  spaces of the manifold, using densities.
Vector-valued {{math|''L<sup>p</sup>''}} spaces
Given a measure space  and a locally-convex space , one may also define a spaces of -integrable E-valued functions in a number of ways.
The most common of these being the spaces of Bochner integrable and Pettis-integrable functions.
Using the tensor product of locally convex spaces, these may be respectively defined as  L^p_{\mu}\left(X,\Sigma,\mu\right)\otimes_{\pi} E  and  L^p_{\mu}\left(X,\Sigma,\mu\right)\otimes_{\epsilon} E ; where  \otimes_{\pi}  and  \otimes_{\epsilon} respectively denote the projective and injective tensor products of locally convex spaces.
When  is a nuclear space, Grothendieck showed that these two constructions are indistinguishable.
See also
\left(\scriptstyle L^1_{\text{loc}}\right)
Notes
References
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External links
Proof that Lp spaces are complete
