L^p space

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 L¹ norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its L² 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 L¹ norm and the L² norm of the parameter vector.

The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps L^p(R) to L^q(R) (or L^p(T) to ℓ^q) respectively, where 1 ≤ p ≤ 2 and 1/p + 1/q = 1 . This is a consequence of the Riesz–Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality.

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

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces L² and ℓ² are both Hilbert spaces. In fact, by choosing a Hilbert basis E, i.e., a maximal orthonormal subset of L² or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to ℓ²(E) (same E as above), i.e., a Hilbert space of type ℓ².

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

$${\displaystyle \left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.}$$

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 L^p-norms for p → ∞. It turns out that this limit is equivalent to the following definition:

$${\displaystyle \left\|x\right\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|\right\}}$$

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 Rⁿ 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, ..., n}.

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:

$${\displaystyle \left\|x\right\|_{2}\leq \left\|x\right\|_{1}.}$$

This fact generalizes to p-norms in that the p-norm ||x||_p of any given vector x does not grow with p:

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

$${\displaystyle \left\|x\right\|_{1}\leq {\sqrt {n}}\left\|x\right\|_{2}~.}$$

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 ℂⁿ where 0 < r < p:

$${\displaystyle \left\|x\right\|_{p}\leq \left\|x\right\|_{r}\leq n^{{\frac {1}{r}}-{\frac {1}{p}}}\left\|x\right\|_{p}~.}$$

This is a consequence of Hölder's inequality.

Astroid, unit circle in p = 2/3 metric

In Rⁿ for n > 1, the formula

$${\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}}$$

defines an absolutely homogeneous function for 0 < p < 1; however, the resulting function does not define a norm, because it is not subadditive. On the other hand, the formula

$${\displaystyle |x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}}$$

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

Hence, the function

$${\displaystyle d_{p}(x,y)=\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}}$$

defines a metric. The metric space (Rⁿ, d_p) is denoted by ℓ_n^p.

Although the p-unit ball B_n^p around the origin in this metric is "concave", the topology defined on Rⁿ by the metric d_p is the usual vector space topology of Rⁿ, hence ℓ_n^p is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of ℓ_n^p is to denote by C_p(n) the smallest constant C such that the multiple C B_n^p of the p-unit ball contains the convex hull of B_n^p, equal to B_n¹. The fact that for fixed p < 1 we have

$${\displaystyle C_{p}(n)=n^{{\frac {1}{p}}-1}\to \infty ,\quad {\text{as }}n\to \infty }$$

shows that the infinite-dimensional sequence space ℓ^p defined below, is no longer locally convex.

There is one ℓ₀ norm and another function called the ℓ₀ "norm" (with quotation marks).

The mathematical definition of the ℓ₀ norm was established by Banach's Theory of Linear Operations. The space of sequences has a complete metric topology provided by the F-norm

$${\displaystyle (x_{n})\mapsto \sum _{n}2^{-n}{\frac {|x_{n}|}{1+|x_{n}|}},}$$

which is discussed by Stefan Rolewicz in Metric Linear Spaces.[1] The ℓ₀-normed space is studied in functional analysis, probability theory, and harmonic analysis.

Another function was called the ℓ₀ "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. Many authors abuse terminology by omitting the quotation marks. Defining 0⁰ = 0, the zero "norm" of x is equal to

$${\displaystyle |x_{1}|^{0}+|x_{2}|^{0}+\cdots +|x_{n}|^{0}.}$$

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 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.

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:

• ℓ¹, the space of sequences whose series is absolutely convergent,
• ℓ², 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:

$${\displaystyle {\begin{aligned}&(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots )+(y_{1},y_{2},\ldots ,y_{n},y_{n+1},\ldots )\\={}&(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n},x_{n+1}+y_{n+1},\ldots ),\\[6pt]&\lambda \cdot \left(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots \right)\\={}&(\lambda x_{1},\lambda x_{2},\ldots ,\lambda x_{n},\lambda x_{n+1},\ldots ).\end{aligned}}}$$

Define the p-norm:

$${\displaystyle \left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}+|x_{n+1}|^{p}+\cdots \right)^{1/p}}$$

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, (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

$${\displaystyle \left(1,{\frac {1}{2}},\ldots ,{\frac {1}{n}},{\frac {1}{n+1}},\ldots \right)}$$

is not in ℓ¹, but it is in ℓ^p for p > 1, as the series

$${\displaystyle 1^{p}+{\frac {1}{2^{p}}}+\cdots +{\frac {1}{n^{p}}}+{\frac {1}{(n+1)^{p}}}+\cdots ,}$$

diverges for p = 1 (the harmonic series), but is convergent for p > 1.

One also defines the ∞-norm using the supremum:

$${\displaystyle \left\|x\right\|_{\infty }=\sup(|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|,|x_{n+1}|,\ldots )}$$

and the corresponding space ℓ^∞ of all bounded sequences. It turns out that[2]

$${\displaystyle \left\|x\right\|_{\infty }=\lim _{p\to \infty }\left\|x\right\|_{p}}$$

if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider ℓ^p spaces for 1 ≤ p ≤ ∞.

The p-norm thus defined on ℓ^p is indeed a norm, and ℓ^p together with this norm is a Banach space. The fully general L^p 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 p-norm.

In complete analogy to the preceding definition one can define the space ${\displaystyle \ell ^{p}(I)}$ over a general index set ${\displaystyle I}$ (and ${\displaystyle 1\leq p<\infty }$) as

$${\displaystyle \ell ^{p}(I)=\left\{(x_{i})_{i\in I}\in \mathbb {K} ^{I}:\sum _{i\in I}|x_{i}|^{p}<+\infty \right\},}$$

where convergence on the right means that only countably many summands are nonzero (see also Unconditional convergence). With the norm

$${\displaystyle \left\|x\right\|_{p}=\left(\sum _{i\in I}|x_{i}|^{p}\right)^{1/p}}$$

the space ${\displaystyle \ell ^{p}(I)}$ becomes a Banach space. In the case where ${\displaystyle I}$ is finite with ${\displaystyle n}$ elements, this construction yields Rⁿ with the ${\displaystyle p}$-norm defined above. If ${\displaystyle I}$ is countably infinite, this is exactly the sequence space ${\displaystyle \ell ^{p}}$ defined above. For uncountable sets ${\displaystyle I}$ this is a non-separable Banach space which can be seen as the locally convex direct limit of ${\displaystyle \ell ^{p}}$-sequence spaces.[3]

For ${\displaystyle p=2,}$ the ${\displaystyle \|\,\cdot \,\|_{2}}$-norm is even induced by a canonical inner product ${\displaystyle \langle \,\cdot ,\,\cdot \rangle ,}$ called the Euclidean inner product, which means that ${\displaystyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}$ holds for all vectors ${\displaystyle \mathbf {x} .}$ This inner product can expressed in terms of the norm by using the polarization identity. On ${\displaystyle \ell ^{2},}$ it can be defined by

$${\displaystyle \langle \left(x_{i}\right)_{i},\left(y_{n}\right)_{i}\rangle _{\ell ^{2}}~=~\sum _{i}x_{i}{\overline {y_{i}}}}$$

while for the space ${\displaystyle L^{2}(X,\mu )}$ associated with a measure space ${\displaystyle (X,\Sigma ,\mu ),}$ which consists of all square-integrable functions, it is

$${\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}f(x){\overline {g(x)}}\,\mathrm {d} x.}$$

Now consider the case ${\displaystyle p=\infty .}$ We can define

$${\displaystyle \ell ^{\infty }(I)=\{x\in \mathbb {K} ^{I}:\sup \operatorname {range} |x|<+\infty \},}$$

[nb 1]

where for all x

$${\displaystyle \|x\|_{\infty }\equiv \inf\{C\in \mathbb {R} _{\geq 0}:|x_{i}|\leq C{\text{ for all }}i\in I\}={\begin{cases}\sup \operatorname {range} |x|&{\text{if }}X\neq \emptyset ,\\0&{\text{if }}X=\emptyset .\end{cases}}}$$

[nb 2][4]

The index set ${\displaystyle I}$ can be turned into a measure space by giving it the discrete σ-algebra and the counting measure. Then the space ${\displaystyle \ell ^{p}(I)}$ is just a special case of the more general ${\displaystyle L^{p}}$-space (see below).

Similar to the ℓ^p spaces, L² is the only Hilbert space among L^p spaces. In the complex case, the inner product on L² is defined by

$${\displaystyle \langle f,g\rangle =\int _{S}f(x){\overline {g(x)}}\,\mathrm {d} \mu (x)}$$

The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in L² 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 S = N, and μ is the counting measure on N. More generally, if one considers any set S with the counting measure, the resulting L^p space is denoted ℓ^p(S). For example, the space ℓ^p(Z) is the space of all sequences indexed by the integers, and when defining the p-norm on such a space, one sums over all the integers. The space ℓ^p(n), where n is the set with n elements, is Rⁿ with its p-norm as defined above. As any Hilbert space, every space L² is linearly isometric to a suitable ℓ²(I), where the cardinality of the set I is the cardinality of an arbitrary Hilbertian basis for this particular L².

The dual space (the Banach space of all continuous linear functionals) of L^p(μ) for 1 < p < ∞ has a natural isomorphism with L^q(μ), where q is such that 1/p + 1/q = 1 (i.e. q = p/p − 1). This isomorphism associates g ∈ L^q(μ) with the functional κ_p(g) ∈ L^p(μ)^∗ defined by

$${\displaystyle f\mapsto \kappa _{p}(g)(f)=\int fg\,\mathrm {d} \mu \ }$$

for every ${\displaystyle f\in L^{p}(\mu )}$

The fact that κ_p(g) is well defined and continuous follows from Hölder's inequality. κ_p : L^q(μ) → L^p(μ)^∗ 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[5]) that any G ∈ L^p(μ)^∗ can be expressed this way: i.e., that κ_p is onto. Since κ_p is onto and isometric, it is an isomorphism of Banach spaces. With this (isometric) isomorphism in mind, it is usual to say simply that L^q is the dual Banach space of L^p.

For 1 < p < ∞, the space L^p(μ) is reflexive. Let κ_p be as above and let κ_q : L^p(μ) → L^q(μ)^∗ be the corresponding linear isometry. Consider the map from L^p(μ) to L^p(μ)^∗∗, obtained by composing κ_q with the transpose (or adjoint) of the inverse of κ_p:

$${\displaystyle j_{p}:L^{p}(\mu )\mathrel {\overset {\kappa _{q}}{\longrightarrow }} L^{q}(\mu )^{*}\mathrel {\overset {\left(\kappa _{p}^{-1}\right)^{*}}{\longrightarrow }} L^{p}(\mu )^{**}}$$

This map coincides with the canonical embedding J of L^p(μ) into its bidual. Moreover, the map j_p is onto, as composition of two onto isometries, and this proves reflexivity.

If the measure μ on S is sigma-finite, then the dual of L¹(μ) is isometrically isomorphic to L^∞(μ) (more precisely, the map κ₁ corresponding to p = 1 is an isometry from L^∞(μ) onto L¹(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 L¹(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 ℓ¹.[6]

Colloquially, if 1 ≤ p < q ≤ ∞, then L^p(S, μ) contains functions that are more locally singular, while elements of L^q(S, μ) can be more spread out. Consider the Lebesgue measure on the half line (0, ∞). A continuous function in L¹ might blow up near 0 but must decay sufficiently fast toward infinity. On the other hand, continuous functions in L^∞ need not decay at all but no blow-up is allowed. The precise technical result is the following.[7] Suppose that 0 < p < q ≤ ∞. Then:

1. L^q(S, μ) ⊂ L^p(S, μ) if and only if S does not contain sets of finite but arbitrarily large measure, and
2. 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 L^p in the first case, and L^p to L^q in the second. (This is a consequence of the closed graph theorem and properties of L^p spaces.) Indeed, if the domain S has finite measure, one can make the following explicit calculation using Hölder's inequality

$${\displaystyle \ \|\mathbf {1} f^{p}\|_{1}\leq \|\mathbf {1} \|_{q/(q-p)}\|f^{p}\|_{q/p}}$$

leading to

$${\displaystyle \ \|f\|_{p}\leq \mu (S)^{1/p-1/q}\|f\|_{q}.}$$

The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity I : L^q(S, μ) → L^p(S, μ) is precisely

$${\displaystyle \|I\|_{q,p}=\mu (S)^{1/p-1/q}}$$

the case of equality being achieved exactly when  f = 1 μ-almost-everywhere.

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

$${\displaystyle f=\sum _{j=1}^{n}a_{j}\mathbf {1} _{A_{j}}}$$

where a_j is scalar, A_j ∈ Σ has finite measure and ${\displaystyle {\mathbf {1} }_{A_{j}}}$ is the indicator function of the set ${\displaystyle A_{j}}$, for j = 1, ..., n. By construction of the integral, the vector space of integrable simple functions is dense in L^p(S, Σ, μ).

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

$${\displaystyle F\subset A\subset U\subset V\quad {\text{and}}\quad \mu (U)-\mu (F)=\mu (U\setminus F)<\varepsilon }$$

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

$${\displaystyle \int _{S}|\mathbf {1} _{A}-\varphi |\,\mathrm {d} \mu <\varepsilon \,.}$$

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 L^p(S, Σ, μ). More precisely, one can use bounded continuous functions that vanish outside one of the open sets V_n.

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(R^d). Similarly, the space of integrable step functions is dense in L^p(R^d); this space is the linear span of indicator functions of bounded intervals when d = 1, of bounded rectangles when d = 2 and more generally of products of bounded intervals.

Several properties of general functions in L^p(R^d) 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 L^p(R^d), in the following sense:

$${\displaystyle \forall f\in L^{p}\left(\mathbf {R} ^{d}\right):\quad \left\|\tau _{t}f-f\right\|_{p}\to 0,\quad {\text{as }}\mathbf {R} ^{d}\ni t\to 0,}$$

where

$${\displaystyle (\tau _{t}f)(x)=f(x-t).}$$

The vector space of (equivalence classes of) measurable functions on (S, Σ, μ) is denoted L⁰(S, Σ, μ) (Kalton, Peck & Roberts 1984). By definition, it contains all the L^p, and is equipped with the topology of convergence in measure. When μ is a probability measure (i.e., μ(S) = 1), this mode of convergence is named convergence in probability.

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

$${\displaystyle V_{\varepsilon }={\Bigl \{}f:\mu {\bigl (}\{x:|f(x)|>\varepsilon \}{\bigr )}<\varepsilon {\Bigr \}},\qquad \varepsilon >0}$$

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

$${\displaystyle d(f,g)=\int _{S}\varphi {\bigl (}|f(x)-g(x)|{\bigr )}\,\mathrm {d} \mu (x)}$$

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 L⁰. Under this metric the space L⁰ is complete (it is again an F-space). The space L⁰ is in general not locally bounded, and not locally convex.

For the infinite Lebesgue measure λ on Rⁿ, the definition of the fundamental system of neighborhoods could be modified as follows

$${\displaystyle W_{\varepsilon }=\left\{f:\lambda \left(\left\{x:|f(x)|>\varepsilon {\text{ and }}|x|<{\frac {1}{\varepsilon }}\right\}\right)<\varepsilon \right\}}$$

The resulting space L⁰(Rⁿ, λ) coincides as topological vector space with L⁰(Rⁿ, g(x) dλ(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

$${\displaystyle \lambda _{f}(t)=\mu \left\{x\in S:|f(x)|>t\right\}}$$

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

$${\displaystyle \lambda _{f}(t)\leq {\frac {\|f\|_{p}^{p}}{t^{p}}}}$$

A function f is said to be in the space weak L^p(S, μ), or L^p,w(S, μ), if there is a constant C > 0 such that, for all t > 0,

$${\displaystyle \lambda _{f}(t)\leq {\frac {C^{p}}{t^{p}}}}$$

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

$${\displaystyle \|f\|_{p,w}=\sup _{t>0}~t\lambda _{f}^{1/p}(t).}$$

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),

$${\displaystyle \|f\|_{p,w}\leq \|f\|_{p}}$$

and in particular L^p(S, μ) ⊂ L^p,w(S, μ).

In fact, one has

$${\displaystyle \|f\|_{L^{p}}^{p}=\int |f(x)|^{p}d\mu (x)\geq \int _{\{|f(x)|>t\}}t^{p}+\int _{\{|f(x)|\leq t\}}|f|^{p}\geq t^{p}\mu (\{|f|>t\}),}$$

and raising to power 1/p and taking the supremum in t one has

$${\displaystyle \|f\|_{L^{p}}\geq \sup _{t>0}t\;\mu (\{|f|>t\})^{1/p}=\|f\|_{L^{p,w}}.}$$

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

For any 0 < r < p the expression

$${\displaystyle |||f|||_{L^{p,\infty }}=\sup _{0<\mu (E)<\infty }\mu (E)^{-1/r+1/p}\left(\int _{E}|f|^{r}\,d\mu \right)^{1/r}}$$

is comparable to the L^p,w-norm. Further in the case p > 1, this expression defines a norm if r = 1. Hence for p > 1 the weak L^p spaces are Banach spaces (Grafakos 2004).

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 L^p(S, w dμ), where w dμ means the measure ν defined by

$${\displaystyle \nu (A)\equiv \int _{A}w(x)\,\mathrm {d} \mu (x),\qquad A\in \Sigma ,}$$

or, in terms of the Radon–Nikodym derivative, w = dν/dμ the norm for L^p(S, w dμ) is explicitly

$${\displaystyle \|u\|_{L^{p}(S,w\,\mathrm {d} \mu )}\equiv \left(\int _{S}w(x)|u(x)|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}}$$

As L^p-spaces, the weighted spaces have nothing special, since L^p(S, w dμ) is equal to L^p(S, dν). But they are the natural framework for several results in harmonic analysis (Grafakos 2004); they appear for example in the Muckenhoupt theorem: for 1 < p < ∞, the classical Hilbert transform is defined on L^p(T, λ) where T denotes the unit circle and λ the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on L^p(Rⁿ, λ). Muckenhoupt's theorem describes weights w such that the Hilbert transform remains bounded on L^p(T, w dλ) and the maximal operator on L^p(Rⁿ, w dλ).

One may also define spaces L^p(M) on a manifold, called the intrinsic L^p spaces of the manifold, using densities.

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 ${\displaystyle L_{\mu }^{p}\left(X,\Sigma ,\mu \right)\otimes _{\pi }E}$ and ${\displaystyle L_{\mu }^{p}\left(X,\Sigma ,\mu \right)\otimes _{\varepsilon }E}$; where ${\displaystyle \otimes _{\pi }}$ and ${\displaystyle \otimes _{\varepsilon }}$ 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.