Lp Space · aiwiki.page

An $L^p$ space is a vector space of measurable functions whose absolute values have an integrable $p$ th power, or are essentially bounded when $p=\infty$ . Functions that agree almost everywhere are treated as the same element. These spaces connect measure theory with functional analysis: for $1\le p\le\infty$ , they are Banach spaces, while $L^2$ also has the structure of a Hilbert space. The exponent specifies how the space measures a function’s size. (math.ucdavis.edu)

Definition and equality almost everywhere

Let $(X,\Sigma,\mu)$ be a measure space, where $\Sigma$ is a sigma-algebra and $\mu$ is a measure. Functions may take values in the real or complex numbers. For $1\le p<\infty$ , define the norm

\|f\|_p=\left(\int_X |f(x)|^p\,d\mu(x)\right)^{1/p}.

The integral is understood as a Lebesgue integral. Then $L^p(X,\Sigma,\mu)$ , often abbreviated $L^p(X)$ or $L^p(\mu)$ , consists of functions for which this quantity is finite, subject to identification almost everywhere. (math.ucdavis.edu)

More precisely, its elements are equivalence classes under

f\sim g\quad\Longleftrightarrow\quad f=g\ \text{almost everywhere}.

This identification is necessary because changing values on a null set does not change the integral. Without it, a nonzero function supported on a null set could have norm zero. For $p=\infty$ ,

\|f\|_\infty=\operatorname*{ess\,sup}_{x\in X}|f(x)|,

where the essential supremum ignores exceptional sets of measure zero. Thus essential boundedness is weaker than boundedness at every point. (math.ucdavis.edu)

Inequalities and completeness

Two inequalities underpin the theory. If $1\le p\le\infty$ and $q$ is its conjugate exponent, meaning

\frac1p+\frac1q=1

with $1/\infty=0$ , Hölder’s inequality states that

\int_X |fg|\,d\mu\le\|f\|_p\|g\|_q.

At $p=q=2$ , this becomes the Cauchy–Schwarz inequality. Minkowski’s inequality gives

\|f+g\|_p\le\|f\|_p+\|g\|_p,

which is the triangle inequality required for a norm. Together with homogeneity and positive definiteness, it makes $L^p$ a normed vector space. (ocw.mit.edu)

Every Cauchy sequence in this norm converges to an element of the same space. This completeness result, commonly called the Riesz–Fischer theorem, explains why $L^p$ spaces are suitable for approximation and limiting arguments. Norm convergence means $\|f_n-f\|_p\to0$ ; it should not be confused with convergence at every point. For finite $p$ , norm convergence guarantees a subsequence converging almost everywhere, but not necessarily almost-everywhere convergence of the full sequence. (ocw.mit.edu)

Examples and dependence on the measure

On a subset of $\mathbb R^n$ , the usual choice is Lebesgue measure. The spaces $L^1$ , $L^2$ , and $L^\infty$ describe absolutely integrable, square-integrable, and essentially bounded functions, respectively. On the positive integers with counting measure, the corresponding spaces are sequence spaces:

\ell^p=\left\{(a_n):\sum_{n=1}^\infty|a_n|^p<\infty\right\}.

Here $\ell^\infty$ consists of bounded sequences. On a finite set with counting measure, the construction gives the familiar finite-dimensional $p$ -norms. (math.ucdavis.edu)

Integrability depends on both the exponent and the underlying measure. For example, on $(0,1)$ ,

f(x)=x^{-\alpha},\qquad \alpha>0,

belongs to $L^p$ exactly when $\alpha p<1$ . Increasing $p$ therefore imposes a stricter condition on this singularity. If $0<\mu(X)<\infty$ and $1\le p<q\le\infty$ , Hölder’s inequality yields

\|f\|_p\le\mu(X)^{1/p-1/q}\|f\|_q,

so $L^q\subseteq L^p$ . On infinite-measure spaces, this inclusion generally fails; by contrast, for counting measure on the integers, $\ell^p\subseteq\ell^q$ when $p<q$ . (math.ucdavis.edu)

Hilbert structure and duality

The space $L^2$ has the inner product

\langle f,g\rangle=\int_X f(x)\overline{g(x)}\,d\mu(x),

which induces its norm. Its completeness makes it a Hilbert space, supporting orthogonality and orthogonal projection. These structures are important in Fourier analysis and approximation. (math.ucdavis.edu)

For a sigma-finite measure space and $1\le p<\infty$ , the continuous dual space of $L^p$ is identified isometrically with $L^q$ , where $q$ is the conjugate exponent. In the real-valued setting, every bounded linear functional has the form

F(f)=\int_X f g\,d\mu,\qquad g\in L^q.

For $1<p<\infty$ , $L^p$ is reflexive. The endpoint $L^\infty$ differs: its continuous dual is generally larger than $L^1$ , so conjugate-exponent duality cannot simply be reversed at that endpoint. (math.ucdavis.edu)

Extensions and applications

The definition also extends to $0<p<1$ . The same integral expression is then a quasi-norm rather than, in general, a norm: the ordinary triangle inequality can fail. Nevertheless,

d(f,g)=\int_X|f-g|^p\,d\mu

defines a complete metric on the almost-everywhere equivalence classes. (math.ucdavis.edu)

Local spaces $L^p_{\mathrm{loc}}(\Omega)$ require integrability on compact subsets rather than on the whole domain. Sobolev spaces add conditions on weak derivatives and are central to the study of partial differential equations. On a probability space, a random variable $Y$ belongs to $L^p$ when its absolute $p$ th moment is finite:

\|Y\|_p=\bigl(\mathbb E|Y|^p\bigr)^{1/p}.

Thus $L^p$ language also expresses moment conditions and mean-power convergence in probability theory. (math.ucdavis.edu)