Lp space

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.

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.

-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

Beyond this qualitative statement, a quantitative way to measure the lack of convexity of

"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

[citation needed] Many authors abuse terminology by omitting the quotation marks.

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.

-norm can be extended to vectors that have an infinite number of components (sequences), which yields the space

Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by:

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,

this is a non-separable Banach space which can be seen as the locally convex direct limit of

-th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified.

forms a vector space when addition and scalar multiplication are defined pointwise.

Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm.

induces a norm (defined shortly) on the canonical quotient vector space of

In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of

For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebra.

This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces

is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous.

It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in

does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.

it is common to work with the Hardy space Hp whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another.

is a well defined continuous linear mapping which is an isometry by the extremal case of Hölder's inequality.

However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo–Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of

As a consequence of the closed graph theorem, the embedding is continuous, i.e., the identity operator is a bounded linear map from

The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity

metrics, and measures of central tendency can be characterized as solutions to variational problems.

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus.

-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.

But they are the natural framework for several results in harmonic analysis (Grafakos 2004); they appear for example in the Muckenhoupt theorem: for