Lp space
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From Wikipedia, the free encyclopedia
In mathematics, the Lp 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).
Lp 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 p-norm in finite dimensions
Illustrations of unit circles (see also superellipse)
in based on different -norms (every vector from the origin to the unit circle
has a length of one, the length being calculated with length-formula of the
corresponding ).
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 (or uniform norm) is the limit of the -norms for
, given by:
For all the -norms and maximum norm satisfy the properties of a "length
function" (or norm), that is:
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 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 p-norms
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 -norms in that the -norm of any given vector does not
grow with :
