In mathematics, and particularly in
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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, Fichera's existence principle is an existence and uniqueness theorem for solution of
functional equations
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
, proved by
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.
Biogra ...
in 1954. More precisely, given a general
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
and two
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
s from it
onto
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
two
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) 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 ...
s, the principle states necessary and sufficient conditions for a
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
between the two
dual Banach spaces to be invertible for every vector in .
[See , , , .]
See also
*
*
*
*
*
Notes
References
*. A survey of Gaetano Fichera's contributions to the theory of partial differential equations, written by two of his pupils.
*.
*.
*: for a review of the book, see .
*. The paper ''Some recent developments of the theory of boundary value problems for linear partial differential equations'' describes Fichera's approach to a general theory of
boundary value problem
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
s for
linear partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ...
s through a theorem similar in spirit to the
Lax–Milgram theorem.
*. A monograph based on lecture notes, taken by
Lucilla Bassotti and
Luciano De Vito of a course held by Gaetano Fichera at the
INdAM: for a review of the book, see .
*.
*, reviewed also by , and by
*.
*.
*. An expository paper detailing the contributions of Gaetano Fichera and his school on the problem of numerical calculation of
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s for general
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
s.
{{Functional analysis
Banach spaces
Normed spaces
Partial differential equations
Theorems in functional analysis