L^p space

In mathematics, the $L p$ 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).

For the sequence space ℓ^p, see Sequence space § ℓp spaces.

$L p$ 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[edit]

Statistics[edit]

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, can be 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 squared $L^{2}$ 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 $L^{1}$ norm and the squared $L^{2}$ norm of the parameter vector.

Hausdorff–Young inequality[edit]

The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps $L^{p}(\mathbb {R} )$ to $L^{q}(\mathbb {R} )$ (or $L^{p}(\mathbf {T} )$ to $\ell ^{q}$ ) respectively, where $1\leq p\leq 2$ and ${\tfrac {1}{p}}+{\tfrac {1}{q}}=1.$ 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 $L^{q}.$

only the zero vector has zero length,

the length of the vector is positive homogeneous with respect to multiplication by a scalar (), and

positive homogeneity

the length of the sum of two vectors is no larger than the sum of lengths of the vectors ().

triangle inequality

$\ell ^{1},$ the space of sequences whose series are ,

absolutely convergent

$\ell ^{2},$ the space of square-summable sequences, which is a , and

Hilbert space

$\ell ^{\infty },$ the space of .

bounded sequences

Generalizations and extensions[edit]

Weak $L p$ [edit]

Let $(S,\Sigma ,\mu )$ be a measure space, and $f$ a measurable function with real or complex values on $S.$ The distribution function of $f$ is defined for $t\geq 0$ by $\lambda _{f}(t)=\mu \{x\in S:|f(x)|>t\}.$

If $f$ is in $L^{p}(S,\mu )$ for some $p$ with $1\leq p<\infty ,$ then by Markov's inequality, $\lambda _{f}(t)\leq {\frac {\|f\|_{p}^{p}}{t^{p}}}$

A function $f$ is said to be in the space weak $L^{p}(S,\mu )$ , or $L^{p,w}(S,\mu ),$ if there is a constant $C>0$ such that, for all $t>0,$ $\lambda _{f}(t)\leq {\frac {C^{p}}{t^{p}}}$

The best constant $C$ for this inequality is the $L^{p,w}$ -norm of $f,$ and is denoted by $\|f\|_{p,w}=\sup _{t>0}~t\lambda _{f}^{1/p}(t).$

The weak $L^{p}$ coincide with the Lorentz spaces $L^{p,\infty },$ 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 $L^{p}(S,\mu ),$ $\|f\|_{p,w}\leq \|f\|_{p}$ and in particular $L^{p}(S,\mu )\subset L^{p,w}(S,\mu ).$

In fact, one has $\|f\|_{L^{p}}^{p}=\int |f(x)|^{p}d\mu (x)\geq \int _{\{|f(x)|>t\}}t^{p}+\int _{\{|f(x)|\leq t\}}|f|^{p}\geq t^{p}\mu (\{|f|>t\}),$ and raising to power $1/p$ and taking the supremum in $t$ one has $\|f\|_{L^{p}}\geq \sup _{t>0}t\;\mu (\{|f|>t\})^{1/p}=\|f\|_{L^{p,w}}.$

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

For any $0<r<p$ 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 $L^{p,w}$ -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.

Weighted $L p$ spaces[edit]

As before, consider a measure space $(S,\Sigma ,\mu ).$ Let $w:S\to [a,\infty ),a>0$ be a measurable function. The $w$ -weighted $L^{p}$ space is defined as $L^{p}(S,w\,\mathrm {d} \mu ),$ where $w\,\mathrm {d} \mu$ means the measure $\nu$ 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, $w={\tfrac {\mathrm {d} \nu }{\mathrm {d} \mu }}$ the norm for $L^{p}(S,w\,\mathrm {d} \mu )$ 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 $L^{p}$ -spaces, the weighted spaces have nothing special, since $L^{p}(S,w\,\mathrm {d} \mu )$ is equal to $L^{p}(S,\mathrm {d} \nu ).$ But they are the natural framework for several results in harmonic analysis (Grafakos 2004); they appear for example in the Muckenhoupt theorem: for $1<p<\infty ,$ the classical Hilbert transform is defined on $L^{p}(\mathbf {T} ,\lambda )$ where $\mathbf {T}$ denotes the unit circle and $\lambda$ the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on $L^{p}(\mathbb {R} ^{n},\lambda ).$ Muckenhoupt's theorem describes weights $w$ such that the Hilbert transform remains bounded on $L^{p}(\mathbf {T} ,w\,\mathrm {d} \lambda )$ and the maximal operator on $L^{p}(\mathbb {R} ^{n},w\,\mathrm {d} \lambda ).$

$L p$ spaces on manifolds[edit]

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

Vector-valued $L p$ spaces[edit]

Given a measure space $(\Omega ,\Sigma ,\mu )$ and a locally convex space $E$ (here assumed to be complete), it is possible to define spaces of $p$ -integrable $E$ -valued functions on $\Omega$ 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 $L^{p}$ topology. Another way involves topological tensor products of $L^{p}(\Omega ,\Sigma ,\mu )$ with $E.$ Element of the vector space $L^{p}(\Omega ,\Sigma ,\mu )\otimes E$ are finite sums of simple tensors $f_{1}\otimes e_{1}+\cdots +f_{n}\otimes e_{n},$ where each simple tensor $f\times e$ may be identified with the function $\Omega \to E$ that sends $x\mapsto ef(x).$ This tensor product $L^{p}(\Omega ,\Sigma ,\mu )\otimes E$ 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 $L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\pi }E,$ and the injective tensor product, denoted by $L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\varepsilon }E.$ In general, neither of these space are complete so their completions are constructed, which are respectively denoted by $L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\pi }E$ and $L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\varepsilon }E$ (this is analogous to how the space of scalar-valued simple functions on $\Omega ,$ when seminormed by any $\|\cdot \|_{p},$ is not complete so a completion is constructed which, after being quotiented by $\ker \|\cdot \|_{p},$ is isometrically isomorphic to the Banach space $L^{p}(\Omega ,\mu )$ ). Alexander Grothendieck showed that when $E$ 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.

Adams, Robert A.; Fournier, John F. (2003), Sobolev Spaces (Second ed.), Academic Press, 978-0-12-044143-3.

ISBN

; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128.

Bahouri, Hajer

(1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-13627-9.

Bourbaki, Nicolas

DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, 3-7643-4231-5.

ISBN

Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience.

Duren, P. (1970), Theory of H^p-Spaces, New York: Academic Press

Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Pearson Education, Inc., pp. 253–257, 0-13-035399-X.

ISBN

Hewitt, Edwin; Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag.

; Peck, N. Tenney; Roberts, James W. (1984), An F-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511662447, ISBN 0-521-27585-7, MR 0808777

Kalton, Nigel J.

(1910), "Untersuchungen über Systeme integrierbarer Funktionen", Mathematische Annalen, 69 (4): 449–497, doi:10.1007/BF01457637, S2CID 120242933

Riesz, Frigyes

(1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.

Rudin, Walter

(1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157

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Titchmarsh, EC

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

L^p space

Applications[edit]

Statistics[edit]

Hausdorff–Young inequality[edit]

positive homogeneity

triangle inequality

absolutely convergent

Hilbert space

bounded sequences

Generalizations and extensions[edit]

Weak $L p$ [edit]

Weighted $L p$ spaces[edit]

$L p$ spaces on manifolds[edit]

Vector-valued $L p$ spaces[edit]

ISBN

Bahouri, Hajer

Bourbaki, Nicolas

ISBN

ISBN

Kalton, Nigel J.

Riesz, Frigyes

Rudin, Walter

Rudin, Walter

Titchmarsh, EC

"Lebesgue space"

Proof that L^p spaces are complete

Lp space

Applications[edit]

Statistics[edit]

Hausdorff–Young inequality[edit]

positive homogeneity

triangle inequality

absolutely convergent

Hilbert space

bounded sequences

Generalizations and extensions[edit]

Weak Lp[edit]

Weighted Lp spaces[edit]

Lp spaces on manifolds[edit]

Vector-valued Lp spaces[edit]

ISBN

Bahouri, Hajer

Bourbaki, Nicolas

ISBN

ISBN

Kalton, Nigel J.

Riesz, Frigyes

Rudin, Walter

Rudin, Walter

Titchmarsh, EC

"Lebesgue space"

Proof that Lp spaces are complete

L^p space

Weak $L p$ [edit]

Weighted $L p$ spaces[edit]

$L p$ spaces on manifolds[edit]

Vector-valued $L p$ spaces[edit]

Proof that L^p spaces are complete