Lp space


In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz.
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.

Preliminaries

The -norm in finite dimensions

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.
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 is the limit of the -norms for, given by:
For all the -norms and maximum norm satisfy the properties of a "length function", that is:
  • only the zero vector has zero length,
  • the length of the vector is positive homogeneous with respect to multiplication by a scalar, and
  • the length of the sum of two vectors is no larger than the sum of lengths of the vectors.
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.

Relations between -norms

The grid distance or rectilinear distance between two points is never shorter than the length of the line segment between them. 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 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

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.

When

There is one norm and another function called the "norm".
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 on the product metric:
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 Many authors abuse terminology by omitting the quotation marks. Defining zero to the power of zero| the zero "norm" of is equal to
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.

spaces and sequence spaces

The -norm can be extended to vectors that have an infinite number of components, which yields the space This contains as special cases:
The space of sequences has a natural vector space structure by applying scalar addition and multiplication. Explicitly, the vector sum and the scalar action for infinite sequences of real 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 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 , but is convergent for
One also defines the -norm using the supremum:
and the corresponding space of all bounded sequences. It turns out that
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.

General ℓ''p''-space

In complete analogy to the preceding definition one can define the space over a general index set as
where convergence on the right requires that only countably many summands are nonzero.
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.
For the -norm is even induced by a canonical inner product called the , which means that holds for all vectors This inner product can be expressed in terms of the norm by using the polarization identity.
On it can be defined by
Now consider the case Define
where for all
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.

''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
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, written, if the set is measurable and has measure zero.
Similarly, a measurable function is by a real number written, if the measurable set has measure zero.
The space is the set of all measurable functions that are bounded almost everywhere and is defined as the infimum of these bounds:
When then this is the same as the essential supremum of the absolute value of :
For example, if is a measurable function that is equal to almost everywhere 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 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 . 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.
That the sum of two -th power integrable functions and is again -th power integrable follows from
although it is also a consequence of Minkowski's inequality
which establishes that satisfies the triangle inequality 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 equal to .
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 does not mention it follows that all have the same zero set. 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 on the canonical quotient vector space of by its vector subspace
This normed quotient space is called 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 , which happens if and only if almost everywhere; if this is the case then and are identified in the quotient space. Hence, strictly speaking consists of equivalence classes of functions.
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.
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 .
The Lebesgue space
The normed vector space is called or the of -th power integrable functions and it is a Banach space for every .
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 then the normed space will be linearly isometrically isomorphic to the normed quotient space via the canonical map ; 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.