In mathematics, the ** L^{p} spaces** are function spaces defined using a natural generalization of the

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of *L*^{p} 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 *L*^{1} norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its *L*^{2} 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 *L*^{1} norm and the *L*^{2} norm of the parameter vector.

The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps *L ^{p}*(

By contrast, if *p* > 2, the Fourier transform does not map into *L ^{q}*.

See also: Square-integrable function |

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces *L*^{2} and *ℓ*^{2} are both Hilbert spaces. In fact, by choosing a Hilbert basis *E*, i.e., a maximal orthonormal subset of *L*^{2} or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to *ℓ*^{2}(*E*) (same *E* as above), i.e., a Hilbert space of type *ℓ*^{2}.

The length of a vector *x* = (*x*_{1}, *x*_{2}, ..., *x _{n}*) in the n-dimensional real vector space

The Euclidean distance between two points x and y is the length ||*x* − *y*||_{2} 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 p-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

For a real number *p* ≥ 1, the **p-norm** or ** L^{p}-norm** of x is defined by

The absolute value bars can be dropped when p is a rational number with an even numerator in its reduced form, and x 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 2-norm, and the 1-norm is the norm that corresponds to the rectilinear distance.

The ** L^{∞}-norm** or maximum norm (or uniform norm) is the limit of the

See *L*-infinity.

For all *p* ≥ 1, the p-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 **R**^{n} together with the p-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 *L ^{p}*-space over {1, 2, ...,

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 p-norms in that the p-norm ||*x*||_{p} of any given vector x does not grow with `p`:

||*x*||_{p+a} ≤ ||*x*||_{p} for any vector x and real numbers *p* ≥ 1 and *a* ≥ 0 . (In fact this remains true for 0 < *p* < 1 and *a* ≥ 0 .)

For the opposite direction, the following relation between the 1-norm and the 2-norm is known:

This inequality depends on the dimension n of the underlying vector space and follows directly from the Cauchy–Schwarz inequality.

In general, for vectors in ℂ^{n} where 0 < *r* < *p*:

This is a consequence of Hölder's inequality.

In **R**^{n} for *n* > 1, the formula

defines an absolutely homogeneous function for 0 <

defines a subadditive function at the cost of losing absolute homogeneity. It does define an F-norm, though, which is homogeneous of degree p.

Hence, the function

defines a metric. The metric space (

Although the p-unit ball *B*_{n}^{p} around the origin in this metric is "concave", the topology defined on **R**^{n} by the metric *d _{p}* is the usual vector space topology of

shows that the infinite-dimensional sequence space

There is one *ℓ*_{0} norm and another function called the *ℓ*_{0} "norm" (with quotation marks).

The mathematical definition of the *ℓ*_{0} 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

Another function was called the *ℓ*_{0} "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 *x*.^{[citation needed]} Many authors abuse terminology by omitting the quotation marks. Defining 0^{0} = 0, the zero "norm" of x is equal to

This is not a norm because it is not homogeneous. For example, scaling the vector *x* 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.

Further information: Sequence space |

The p-norm can be extended to vectors that have an infinite number of components (sequences), which yields the space *ℓ*^{p}. This contains as special cases:

*ℓ*^{1}, the space of sequences whose series is absolutely convergent,*ℓ*^{2}, the space of**square-summable**sequences, which is a Hilbert space, and*ℓ*^{∞}, 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:

Define the p-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, (1, 1, 1, ...), will have an infinite p-norm for 1 ≤ *p* < ∞. The space *ℓ ^{p}* is then defined as the set of all infinite sequences of real (or complex) numbers such that the p-norm is finite.

One can check that as p increases, the set *ℓ ^{p}* grows larger. For example, the sequence

is not in

diverges for

One also defines the ∞-norm using the supremum:

and the corresponding space

if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider

The *p*-norm thus defined on *ℓ ^{p}* is indeed a norm, and

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

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 We can define

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 (see below).

An *L ^{p}* 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 1 ≤

The set of such functions forms a vector space, with the following natural operations:

for every scalar

That the sum of two *p*-th power integrable functions is again *p*-th power integrable follows from the inequality

(This comes from the convexity of for .)

In fact, more is true. *Minkowski's inequality* says the triangle inequality holds for || · ||_{p}. Thus the set of *p*-th power integrable functions, together with the function || · ||_{p}, is a seminormed vector space, which is denoted by .

For *p* = ∞, the space is the space of measurable functions bounded almost everywhere, with (when μ(X)≠0) the essential supremum of its absolute value as a norm:

As in the discrete case, if there exists *q* < ∞ such that *f* ∈ *L*^{∞}(*S*, *μ*) ∩ *L ^{q}*(

can be made into a normed vector space in a standard way; one simply takes the quotient space with respect to the subspace of functions whose p-norm is zero. Since for any measurable function *f* , we have that || *f* ||_{p} = 0 if and only if *f* = 0 almost everywhere, that subspace does not depend upon p,

In the quotient space, two functions *f* and g are identified if *f* = *g* almost everywhere. The resulting normed vector space is, by definition,

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.

When the underlying measure space S is understood, *L ^{p}*(

For 1 ≤ *p* ≤ ∞, *L ^{p}*(

The above definitions generalize to Bochner spaces.

Similar to the *ℓ*^{p} spaces, *L*^{2} is the only Hilbert space among *L ^{p}* spaces. In the complex case, the inner product on

The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in *L*^{2} 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 *L*^{∞} 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 *L*^{∞} defines a bounded operator on any *L ^{p}* space by multiplication.

For 1 ≤ *p* ≤ ∞ the ℓ^{p} spaces are a special case of *L ^{p}* spaces, when

The dual space (the Banach space of all continuous linear functionals) of *L ^{p}*(

for every

The fact that *κ _{p}*(

For 1 < *p* < ∞, the space *L ^{p}*(

This map coincides with the canonical embedding J of *L ^{p}*(

If the measure μ on S is sigma-finite, then the dual of *L*^{1}(*μ*) is isometrically isomorphic to *L*^{∞}(*μ*) (more precisely, the map *κ*_{1} corresponding to *p* = 1 is an isometry from *L*^{∞}(*μ*) onto *L*^{1}(*μ*)^{∗}).

The dual of *L*^{∞} is subtler. Elements of *L*^{∞}(*μ*)^{∗} can be identified with bounded signed *finitely* additive measures on S that are absolutely continuous with respect to μ. See ba space for more details. If we assume the axiom of choice, this space is much bigger than *L*^{1}(*μ*) 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 *ℓ*^{∞} is *ℓ*^{1}.^{[6]}

Colloquially, if 1 ≤ *p* < *q* ≤ ∞, then *L ^{p}*(

*L*(^{q}*S*,*μ*) ⊆*L*(^{p}*S*,*μ*) if and only if S does not contain sets of finite but arbitrarily large measure, and*L*(^{p}*S*,*μ*) ⊆*L*(^{q}*S*,*μ*) if and only if S 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
*L ^{q}* to

leading to

The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity *I* : *L ^{q}*(

the case of equality being achieved exactly when

Throughout this section we assume that: 1 ≤ *p* < ∞.

Let (*S*, Σ, *μ*) be a measure space. An *integrable simple function* *f* on S is one of the form

where

More can be said when S is a normal topological space and Σ its Borel σ–algebra, i.e., the smallest σ–algebra of subsets of S containing the open sets.

Suppose *V* ⊂ *S* is an open set with *μ*(*V*) < ∞. It can be proved that for every Borel set *A* ∈ Σ contained in V, and for every *ε* > 0, there exist a closed set F and an open set U such that

It follows that there exists a continuous Urysohn function 0 ≤ *φ* ≤ 1 on *S* that is 1 on *F* and 0 on *S* ∖ *U*, with

If S can be covered by an increasing sequence (*V _{n}*) of open sets that have finite measure, then the space of p–integrable continuous functions is dense in

This applies in particular when *S* = **R**^{d} and when μ is the Lebesgue measure. The space of continuous and compactly supported functions is dense in *L ^{p}*(

Several properties of general functions in *L ^{p}*(

where

Let (*S*, Σ, *μ*) be a measure space. If 0 < *p* < 1, then *L ^{p}*(

As before, we may introduce the p-norm || *f* ||_{p} = *N _{p}*(

and so the function

is a metric on

In this setting *L ^{p}* satisfies a

This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces *L ^{p}* for 1 <

The space *L ^{p}* for 0 <

The only nonempty convex open set in *L ^{p}*([0, 1]) is the entire space (Rudin 1991, §1.47). As a particular consequence, there are no nonzero linear functionals on

The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on **R**^{n}, rather than work with *L ^{p}* for 0 <

The vector space of (equivalence classes of) measurable functions on (*S*, Σ, *μ*) is denoted *L*^{0}(*S*, Σ, *μ*) (Kalton, Peck & Roberts 1984). By definition, it contains all the *L ^{p}*, and is equipped with the topology of

The description is easier when μ is finite. If μ is a finite measure on (*S*, Σ), the 0 function admits for the convergence in measure the following fundamental system of neighborhoods

The topology can be defined by any metric d of the form

where φ is bounded continuous concave and non-decreasing on [0, ∞), with *φ*(0) = 0 and *φ*(*t*) > 0 when *t* > 0 (for example, *φ*(*t*) = min(*t*, 1)). Such a metric is called Lévy-metric for *L*^{0}. Under this metric the space *L*^{0} is complete (it is again an F-space). The space *L*^{0} is in general not locally bounded, and not locally convex.

For the infinite Lebesgue measure λ on **R**^{n}, the definition of the fundamental system of neighborhoods could be modified as follows

The resulting space *L*^{0}(**R**^{n}, *λ*) coincides as topological vector space with *L*^{0}(**R**^{n}, *g*(*x*) d*λ*(x)), for any positive λ–integrable density g.

Let (*S*, Σ, *μ*) be a measure space, and *f* a measurable function with real or complex values on *S*. The distribution function of *f* is defined for *t* ≥ 0 by

If *f* is in *L*^{p}(*S*, *μ*) for some *p* with 1 ≤ *p* < ∞, then by Markov's inequality,

A function *f* is said to be in the space **weak L^{p}(S, μ)**, or

The best constant *C* for this inequality is the *L*^{p,w}-norm of *f*, and is denoted by

The weak *L*^{p} coincide with the Lorentz spaces *L*^{p,∞}, so this notation is also used to denote them.

The *L*^{p,w}-norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for *f* in *L*^{p}(*S*, *μ*),

and in particular

In fact, one has

and raising to power 1/

Under the convention that two functions are equal if they are equal *μ* almost everywhere, then the spaces *L*^{p,w} are complete (Grafakos 2004).

For any 0 < *r* < *p* the expression

is comparable to the

A major result that uses the *L*^{p,w}-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.

As before, consider a measure space (*S*, Σ, *μ*). Let *w* : *S* → [a, ∞), a > 0 be a measurable function. The w-**weighted L^{p} space** is defined as

or, in terms of the Radon–Nikodym derivative, *w* = d*ν*/d*μ* the norm for *L ^{p}*(

As *L ^{p}*-spaces, the weighted spaces have nothing special, since

One may also define spaces *L ^{p}*(

Given a measure space (*X*, Σ, *μ*) and a locally-convex space *E*, one may also define a spaces of *p*-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 and ; where and respectively denote the projective and injective tensor products of locally convex spaces. When *E* is a nuclear space, Grothendieck showed that these two constructions are indistinguishable.