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). L^p spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.

The length of a vector x = (x₁, x₂, ..., x_n) in the n-dimensional real vector space Rⁿ is usually given by the Euclidean norm:

http://upload.wikimedia.org/math/9/c/4/9c4973edb3ed01bdad2ee7e22fac21b4.png

The Euclidean distance between two points x and yis the length ||x − y||₂ of the straight line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space. For example, taxi drivers in Manhattan should measure distance not in terms of the length of the straight line to their destination, but in terms of the Manhattan distance, which takes into account that streets are either orthogonal or parallel to each other. The class of p-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

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

http://upload.wikimedia.org/math/b/7/0/b7022490915ed7618b8265e45cea1df1.png

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 Manhattan distance.【见下面有关于L1-Norm的例子】

The L^∞-norm or maximum norm (or uniform norm) is the limit of the L^p-norms forp → ∞. It turns out that this limit is equivalent to the following definition:

http://upload.wikimedia.org/math/2/0/b/20bfa1315052fef5350bc630321920a1.png

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, 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 Banach space. This Banach space is theL^p-space over Rⁿ.

Relations between p-norms[edit]

The grid distance ("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:

http://upload.wikimedia.org/math/1/f/c/1fc7544fca68ecf8be015c528f2fec61.png

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

||x||_p+a ≤ ||x||_p for any vector x and real numbers p ≥ 1 and a ≥ 0. (In fact this remains true for 0 < p < 1 anda ≥ 0.)

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

http://upload.wikimedia.org/math/d/8/9/d89984599da2908e335acb18d9d9279f.png

This inequality depends on the dimension n of the underlying vector space and follows directly from theCauchy–Schwarz inequality. (柯西-施瓦茨不等式)

In general, for vectors in Cⁿ where 0 < r < p:

http://upload.wikimedia.org/math/b/e/a/bea4bafa8ea945caaa7017cf295c8681.png

When 0 < p < 1

=======================================

L1- Norms

http://en.wikipedia.org/wiki/Taxicab_geometry

Taxicab geometry, considered by Hermann Minkowski in 19th century Germany, is a form of geometry in which the usual distance function or metric ofEuclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, L₁distance or http://upload.wikimedia.org/math/6/c/7/6c72c4de2714433908be059a006a95d9.png norm (see L^p space), city block distance, Manhattan distance, or Manhattan length, with corresponding variations in the name of the geometry.^[1] The latter names allude to the grid layout of most streets on the island of Manhattan, which causes the shortest path a car could take between two intersections in the borough to have length equal to the intersections' distance in taxicab geometry.

http://upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/283px-Manhattan_distance.svg.png

Figure illustrating Manhattan versus Euclidean distance. The red, blue, and yellow lines all have the same length (12), whereas the green line has length http://upload.wikimedia.org/math/5/1/5/515b0c3761f1dcebfc5f9f17f1dd7564.png.

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柯西-施瓦茨不等式

http://en.wikipedia.org/wiki/Cauchy–Schwarz_inequality

柯西-施瓦茨不等式叙述，对于一个内积空间所有向量x和y，

http://upload.wikimedia.org/math/8/3/2/832db451ecd9f5c2b53bab5eeb8f4ce7.png。

其中http://upload.wikimedia.org/math/e/0/2/e02eaeb6eb365f078ca029f67f7a6973.png表示内积，也叫点积。等价地，将两边开方，引用向量的范数，不等式可写为

http://upload.wikimedia.org/math/7/3/6/736808b302d593da008f47d29ccdb6f3.png

另外，等式成立当且仅当x和y线性相关（或者在几何上，它们是平行的，或其中一个向量的模为0）。

若http://upload.wikimedia.org/math/b/b/e/bbef11fa5e0859cc2553bc77f4ea4da9.png有虚部，内积即为标准内积，用拔标记共轭复数那么这个不等式可以更明确的表述为

http://upload.wikimedia.org/math/0/6/b/06ba55090559b51a6990503b913d3d61.png

柯西—施瓦茨不等式的一个重要结果，是内积为连续函数，甚至是满足1阶利普希茨条件的函数。

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