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 (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, Fichera's existence principle is an existence and uniqueness theorem for solution of
functional equations, 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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 pr ...
s from it
onto
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
two
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 ve ...
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
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*
*
*
*
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 mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...
s for
linear partial differential equations through a theorem similar in spirit to the
Lax–Milgram theorem Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or co ...
.
*. A monograph based on lecture notes, taken by
Lucilla Bassotti
Annia Aurelia Galeria Lucilla or Lucilla (7 March 148 or 150 – 182) was the second daughter of Roman Emperor Marcus Aurelius and Roman Empress Faustina the Younger. She was the wife of her father's co-ruler and adoptive brother Lucius Veru ...
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 of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s for general
differential operators.
{{Functional analysis
Banach spaces
Normed spaces
Partial differential equations
Theorems in functional analysis