The basic theorems of functional analysis are Hanbanach theorem, axiom of choice's lemma and Zon's contraction mapping theorem, which are weaker than Brunswick's ideal theorem.
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Functional analysis was formed in 1930s. It is developed from the study of variational method, differential equation, integral equation, function theory and quantum physics. It can be regarded as infinite dimensional analysis by using the viewpoints and methods of geometry and algebra. For more than half a century, on the one hand, it has continuously extracted its research objects and some research methods from the materials provided by many other disciplines, and formed many important branches of its own, such as operator spectrum theory, Banach algebra, topological linear space (also known as topological vector space) theory, generalized function theory and so on.
On the other hand, it has also effectively promoted the development of many other analytical disciplines. It has important applications in differential equations, probability theory, function theory, continuum mechanics, quantum physics, computational mathematics, cybernetics, optimization theory and other disciplines. It is also a basic tool for establishing harmonic analysis theory on groups and one of the important and natural tools for studying infinite freedom physical systems. Today, its viewpoints and methods have penetrated into many engineering disciplines and become one of the foundations of modern analysis.
Baidu Encyclopedia-Functional Analysis