Some math definitions for variational calculus

source: Wikipedia

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,

Hilbert spaces

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces L2 and ℓ2 are both Hilbert spaces. In fact, by choosing a Hilbert basis (i.e., a maximal orthonormal subset of L2 or any Hilbert space), one sees that all Hilbert spaces are isometric to ℓ2(E), where E is a set with an appropriate cardinality.

a locally integrable function (sometimes also called locally summable function)^[1] is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to L_p spaces, but its members are not required to satisfy any growth restriction on their behavior at infinity: in other words, locally integrable functions can grow arbitrarily fast at infinity, but are still manageable in a way similar to ordinary integrable functions.

Standard definition

Definition 1.^[2] Let Ω be an open set in the Euclidean space ℝⁿ and f : Ω → ℂ be a Lebesgue measurable function. If f on Ω is such that

i.e. its Lebesgue integral is finite on all compact subsets K of Ω,^[3] then f  is called locally integrable. The set of all such functions is denoted by L_1,loc(Ω):

where f |_K denotes the restriction of f  to the set K.

The classical definition of a locally integrable function involves only measure theoretic and topological^[4] concepts and can be carried over abstract to complex-valued functions on a topological measure space (X, Σ, μ):^[5] however, since the most common application of such functions is to distribution theory on Euclidean spaces,^[2] all the definitions in this and the following sections deal explicitly only with this important case.

a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

A Banach space is a vector space X over the field R of real numbers, or over the field C of complex numbers, which is equipped with a norm and which is complete with respect to that norm, that is to say, for every Cauchy sequence {x_n} in X, there exists an element x in X such that

or equivalently:

The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space X is a Banach space if and only if each absolutely convergent series in X converges,^[2]

Completeness of a normed space is preserved if the given norm is replaced by an equivalent one.

All norms on a finite-dimensional vector space are equivalent. Every finite-dimensional normed space over R or C is a Banach space.^[3]