In functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, a branch of mathematics, a Beppo-Levi space, named after Beppo Levi

Beppo Levi

Beppo Levi was an Italian mathematician. He published high-level academic articles and books not only in mathematics, but also in physics, history, philosophy, and pedagogy. Levi was a member of the Bologna Academy of Sciences and of the Accademia dei Lincei.- Early years :Beppo Levi was born on...

, is a certain space of generalized function

Generalized function

In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...

s.

In the following, is the space of distributions

Distribution (mathematics)

In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

, is the space of tempered distributions in , the differentiation operator with a multi-index, and is the Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 of .

The Beppo-Levi space is


( denotes the Sobolev

Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...

semi-norm)

An alternative definition is:

Given such that

Let

Then denotes the Beppo-Levi space