In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910).

Lp 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, economics, finance, engineering, and other disciplines.

Applications

Statistics

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, can be 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 squared norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage sparse solutions (where the many parameters are zero).[1] Elastic net regularization uses a penalty term that is a combination of the norm and the squared 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

See also: Square-integrable function

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 p-norm in finite dimensions

Illustrations of unit circles (see also superellipse) in based on different -norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding ).

The euclidean length of a vector in the -dimensional real vector space is given by the Euclidean norm:

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 appropriate for capturing the actual distances in a given space. In contrast, consider 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

The absolute value bars can be dropped when is a rational number with an even numerator in its reduced form, and is drawn from the set of real numbers, or one of its subsets.

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:

See L-infinity.

For all the -norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or norm), which are that:

Abstractly speaking, this means that together with the -norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space. This Banach space is the -space over

Relations between 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:

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:

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

This is a consequence of Hölder's inequality.

When 0 < p < 1

Astroid, unit circle in metric

In for the formula

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
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

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 scalar multiple of the -unit ball contains the convex hull of which is equal to The fact that for fixed we have

shows that the infinite-dimensional sequence space defined below, is no longer locally convex.[citation needed]

When 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

which is discussed by Stefan Rolewicz in Metric Linear Spaces.[2] 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 [citation needed] Many authors abuse terminology by omitting the quotation marks. Defining the zero "norm" of is equal to

An animated gif of p-norms 0.1 through 2 with a step of 0.05.
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 p-norm in infinite dimensions and p spaces

The sequence space p

Further information: Sequence space

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 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:

Define the -norm:

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

is not in but it is in for as the series
diverges for (the harmonic series), but is convergent for

One also defines the -norm using the supremum:

and the corresponding space of all bounded sequences. It turns out that[3]
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 ℓp-space

In complete analogy to the preceding definition one can define the space over a general index set (and ) as

where convergence on the right means that only countably many summands are nonzero (see also Unconditional convergence). With the norm
the space becomes a Banach space. In the case where is finite with elements, this construction yields with the -norm defined above. If is countably infinite, this is exactly the sequence space defined above. For uncountable sets this is a non-separable Banach space which can be seen as the locally convex direct limit of -sequence spaces.[4]

For the -norm is even induced by a canonical inner product called the Euclidean inner product, which means that holds for all vectors This inner product can expressed in terms of the norm by using the polarization identity. On it can be defined by

while for the space associated with a measure space which consists of all square-integrable functions, it is

Now consider the case Define[note 1]

where for all [5][note 2]

The index set can be turned into a measure space by giving it the discrete σ-algebra and the counting measure. Then the space is just a special case of the more general -space (defined below).

Lp spaces and Lebesgue integrals

An space may be defined as a space of measurable functions for which the -th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let be a measure space and [note 3] When , consider the set of all measurable functions from to or whose absolute value raised to the -th power has a finite integral, or in symbols:

To define the set for recall that two functions and defined on are said to be equal almost everywhere, written a.e., if the set is measurable and has measure zero. Similarly, a measurable function (and its absolute value) is bounded (or dominated) almost everywhere by a real number written a.e., if the (necessarily) measurable set has measure zero. The space is the set of all measurable functions that are bounded almost everywhere (by some real ) and is defined as the infimum of these bounds:

When then this is the same as the essential supremum of the absolute value of :[note 4]

For example, if is a measurable function that is equal to almost everywhere[note 5] then for every and thus for all

For every positive the value under of a measurable function and its absolute value are always the same (that is, for all ) and so a measurable function belongs to if and only if its absolute value does. Because of this, many formulas involving -norms are stated only for non-negative real-valued functions. Consider for example the identity which holds whenever is measurable, is real, and (here when ). The non-negativity requirement can be removed by substituting in for which gives Note in particular that when is finite then the formula relates the -norm to the -norm.

Seminormed space of -th power integrable functions

Each set of functions forms a vector space when addition and scalar multiplication are defined pointwise.[note 6] That the sum of two -th power integrable functions and is again -th power integrable follows from [proof 1] although it is also a consequence of Minkowski's inequality

which establishes that satisfies the triangle inequality for (the triangle inequality does not hold for ). That is closed under scalar multiplication is due to being absolutely homogeneous, which means that for every scalar and every function

Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm. Thus is a seminorm and the set of -th power integrable functions together with the function defines a seminormed vector space. In general, the seminorm is not a norm because there might exist measurable functions that satisfy but are not identically equal to [note 5] ( is a norm if and only if no such exists).

Zero sets of -seminorms

If is measurable and equals a.e. then for all positive On the other hand, if is a measurable function for which there exists some such that then almost everywhere. When is finite then this follows from the case and the formula mentioned above.

Thus if is positive and is any measurable function, then if and only if almost everywhere. Since the right hand side ( a.e.) does not mention it follows that all have the same zero set (it does not depend on ). So denote this common set by

This set is a vector subspace of for every positive

Quotient vector space

Like every seminorm, the seminorm induces a norm (defined shortly) on the canonical quotient vector space of by its vector subspace This normed quotient space is called Lebesgue space and it is the subject of this article. We begin by defining the quotient vector space.

Given any the coset consists of all measurable functions that are equal to almost everywhere. The set of all cosets, typically denoted by

forms a vector space with origin when vector addition and scalar multiplication are defined by and This particular quotient vector space will be denoted by

Two cosets are equal if and only if (or equivalently, ), which happens if and only if almost everywhere; if this is the case then and are identified in the quotient space.

The -norm on the quotient vector space

Given any the value of the seminorm on the coset is constant and equal to denote this unique value by so that:

This assignment defines a map, which will also be denoted by on the quotient vector space
This map is a norm on called the -norm. The value of a coset is independent of the particular function that was chosen to represent the coset, meaning that if is any coset then for every (since for every ).

The Lebesgue space

The normed vector space is called space or the Lebesgue space of -th power integrable functions and it is a Banach space for every (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem). When the underlying measure space is understood then is often abbreviated or even just Depending on the author, the subscript notation might denote either or

If the seminorm on happens to be a norm (which happens if and only if ) then the normed space will be linearly isometrically isomorphic to the normed quotient space via the canonical map (since ); in other words, they will be, up to a linear isometry, the same normed space and so they may both be called " space".

The above definitions generalize to Bochner spaces.

In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of in For however, there is a theory of lifts enabling such recovery.

Special cases

Similar to the spaces, is the only Hilbert space among spaces. In the complex case, the inner product on is defined by

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 square-integrable functions, 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 (Titchmarsh 1976).

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 consists of the natural numbers 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 Lp spaces

As in the discrete case, if there exists such that then

Hölder's inequality

Suppose satisfy (where ). If and then and[6]

This inequality, called Hölder's inequality, is in some sense optimal[6] since if (so ) and is a measurable function such that

where the supremum is taken over the closed unit ball of then and

Minkowski inequality

Minkowski inequality, which states that satisfies the triangle inequality, can be generalized: If the measurable function is non-negative (where and are measure spaces) then for all [7]

Atomic decomposition

If then every non-negative has an atomic decomposition,[8] meaning that there exist a sequence of non-negative real numbers and a sequence of non-negative functions called the atoms, whose supports are pairwise disjoint sets of measure such that

and for every integer
and
and where moreover, the sequence of functions depends only on (it is independent of ).[8] These inequalities guarantee that for all integers while the supports of being pairwise disjoint implies[8]

An atomic decomposition can be explicitly given by first defining for every integer [8]

(this infimum is attained by that is, holds) and then letting
where denotes the measure of the set and denotes the indicator function of the set The sequence is decreasing and converges to as [8] Consequently, if then and so that is identically equal to (in particular, the division by causes no issues).

The complementary cumulative distribution function of that was used to define the also appears in the definition of the weak -norm (given below) and can be used to express the -norm (for ) of as the integral[8]

where the integration is with respect to the usual Lebesgue measure on

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

for every

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[9]) 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 continuous dual 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

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 [10]

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.[11] Suppose that Then:

  1. if and only if does not contain sets of finite but arbitrarily large measure (any finite measure, for example).
  2. if and only if does not contain sets of non-zero but arbitrarily small measure (the counting measure, for example).

Neither condition holds for the real line with the Lebesgue measure while both conditions holds for the counting measure on any finite set. 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

leading to

The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity is precisely

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

where are scalars, has finite measure and is the indicator function of the set 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 𝜎–algebra, i.e., the smallest 𝜎–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

It follows that there exists a continuous Urysohn function on that is on and on with

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:

where

Closed subspaces

If is any positive real number, is a probability measure on a measurable space (so that ), and is a vector subspace, then is a closed subspace of if and only if is finite-dimensional[12] ( was chosen independent of ). In this theorem, which is due to Alexander Grothendieck,[12] it is crucial that the vector space be a subset of since it is possible to construct an infinite-dimensional closed vector subspace of (that is even a subset of ), where is Lebesgue measure on the unit circle and is the probability measure that results from dividing it by its mass [12]

Lp (0 < p < 1)

Let be a measure space. If then can be defined as above: it is the quotient vector space of those measurable functions such that

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 (Rudin 1991, §1.47)

and so the function
is a metric on The resulting metric space is complete;[13] the verification is similar to the familiar case when The balls
form a local base at the origin for this topology, as ranges over the positive reals.[13] These balls satisfy for all real which in particular shows that is a bounded neighborhood of the origin;[13] in other words, this space is locally bounded, just like every normed space, despite not being a norm.

In this setting satisfies a reverse Minkowski inequality, that is for

This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces for (Adams & Fournier 2003).

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 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 (Rudin 1991, §1.47). As a particular consequence, there are no nonzero continuous linear functionals on the continuous 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 Hp 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 Hp for (Duren 1970, §7.5).

L0, the space of measurable functions

The vector space of (equivalence classes of) measurable functions on is denoted (Kalton, Peck & Roberts 1984). 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 space is always a topological abelian group but is only a topological vector space if This is because scalar multiplication is continuous if and only if If is -finite then the weaker topology of local convergence in measure is an F-space, i.e. a completely metrizable topological vector space. Moreover, this topology is isometric to global convergence in measure for a suitable choice of probability measure

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

The topology can be defined by any metric of the form

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. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if . To see this, consider the Lebesgue measurable function defined by . Then clearly . 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

The resulting space , with the topology of local convergence in measure, is isomorphic to the space for any positive –integrable density

Generalizations and extensions

Weak Lp

Let be a measure space, and a measurable function with real or complex values on The distribution function of is defined for by

If is in for some with then by Markov's inequality,

A function is said to be in the space weak , or if there is a constant such that, for all

The best constant for this inequality is the -norm of and is denoted by

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

and in particular

In fact, one has

and raising to power and taking the supremum in one has

Under the convention that two functions are equal if they are equal almost everywhere, then the spaces are complete (Grafakos 2004).

For any the expression

is comparable to the -norm. Further in the case this expression defines a norm if Hence for the weak spaces are Banach spaces (Grafakos 2004).

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 Lp spaces

As before, consider a measure space Let be a measurable function. The -weighted space is defined as where means the measure defined by

or, in terms of the Radon–Nikodym derivative, the norm for is explicitly

As -spaces, the weighted spaces have nothing special, since is equal to But they are the natural framework for several results in harmonic analysis (Grafakos 2004); 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

Lp spaces on manifolds

One may also define spaces on a manifold, called the intrinsic spaces of the manifold, using densities.

Vector-valued Lp spaces

Given a measure space and a locally convex space (here assumed to be complete), it is possible to define spaces of -integrable -valued functions on in a number of ways. One way is to define the spaces of Bochner integrable and Pettis integrable functions, and then endow them with locally convex TVS-topologies that are (each in their own way) a natural generalization of the usual topology. Another way involves topological tensor products of with Element of the vector space are finite sums of simple tensors where each simple tensor may be identified with the function that sends This tensor product is then endowed with a locally convex topology that turns it into a topological tensor product, the most common of which are the projective tensor product, denoted by and the injective tensor product, denoted by In general, neither of these space are complete so their completions are constructed, which are respectively denoted by and (this is analogous to how the space of scalar-valued simple functions on when seminormed by any is not complete so a completion is constructed which, after being quotiented by is isometrically isomorphic to the Banach space ). Alexander Grothendieck showed that when is a nuclear space (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.

See also

Notes

  1. ^ Hastie, T.J., Tibshirani, R., & Wainwright, M.J. (2015). Statistical Learning with Sparsity: The Lasso and Generalizations.
  2. ^ Rolewicz, Stefan (1987), Functional analysis and control theory: Linear systems, Mathematics and its Applications (East European Series), vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi+524, doi:10.1007/978-94-015-7758-8, ISBN 90-277-2186-6, MR 0920371, OCLC 13064804[page needed]
  3. ^ Maddox, I. J. (1988), Elements of Functional Analysis (2nd ed.), Cambridge: CUP, page 16
  4. ^ 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
  5. ^ Garling, D. J. H. (2007). Inequalities: A Journey into Linear Analysis. Cambridge University Press. p. 54. ISBN 978-0-521-87624-7.
  6. ^ a b Bahouri, Chemin & Danchin 2011, pp. 1–4.
  7. ^ Bahouri, Chemin & Danchin 2011, p. 4.
  8. ^ a b c d e f Bahouri, Chemin & Danchin 2011, pp. 7–8.
  9. ^ Rudin, Walter (1980), Real and Complex Analysis (2nd ed.), New Delhi: Tata McGraw-Hill, ISBN 9780070542341, Theorem 6.16
  10. ^ Schechter, Eric (1997), Handbook of Analysis and its Foundations, London: Academic Press Inc. See Sections 14.77 and 27.44–47
  11. ^ Villani, Alfonso (1985), "Another note on the inclusion Lp(μ) ⊂ Lq(μ)", Amer. Math. Monthly, 92 (7): 485–487, doi:10.2307/2322503, JSTOR 2322503, MR 0801221
  12. ^ a b c Rudin 1991, pp. 117–119.
  13. ^ a b c Rudin 1991, p. 37.
  1. ^ The condition is not equivalent to being finite, unless
  2. ^ If then
  3. ^ The definitions of and can be extended to all (rather than just ), but it is only when that is guaranteed to be a norm (although is a quasi-seminorm for all ).
  4. ^ If then
  5. ^ a b For example, if a non-empty measurable set of measure exists then its indicator function satisfies although
  6. ^ Explicitly, the vector space operations are defined by:
    for all and all scalars These operations make