## functional analysis

In mathematical physics, constructive quantum field theory is the field devoted to attempts to put quantum field theory on a basis of completely defined concepts from functional analysis. It is known that a quantum field is inherently hard to handle using conventional mathematical techniques like explicit estimates. This is because a quantum field has the general nature of a operator-valued distribution, a type of object from mathematical analysis. This implies that existence theorems for quantum fields can be expected to be very difficult to find, if indeed they are possible at all. ...more on Wikipedia about "Constructive quantum field theory"

In mathematics, the continuous functional calculus of operator theory and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. More precisely, ...more on Wikipedia about "Continuous functional calculus"

In functional analysis, it is often convenient to define something on a normed vector space by defining it on a dense set and extending it to the whole space. This procedure is justified for bounded linear operators by the theorem below. The result is again linear and bounded (and thus continuous), so it is called the continuous linear extension. ...more on Wikipedia about "Continuous linear extension"

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ...more on Wikipedia about "Continuous linear operator"

In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytic expressions, as functions either of time or of frequency. ...more on Wikipedia about "Continuous wavelet"

In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval. ...more on Wikipedia about "Convolution"

A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. The formalism was introduced by John von Neumann (according to other sources independently by Lev Landau and Felix Bloch) in 1927. It is the ...more on Wikipedia about "Density matrix" It's time to think about www.shortopedia.com. functional_analysis

In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one the papers in the series On Rings of Operators. One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to to prove a continuous analogue of the Artin-Wedderburn theorem classifying semi-simple rings. ...more on Wikipedia about "Direct integral"

The non-trivial case is when X is infinite-dimensional and Y is nonzero, and it turns out that in this case one can always find a discontinuous linear map from X to Y. ...more on Wikipedia about "Discontinuous linear map"

In function theory, the disk algebra is the set of continuous functions ...more on Wikipedia about "Disk algebra"

In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form. ...more on Wikipedia about "Dual pair"

In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). The construction can also take place for infinite-dimensional spaces and gives rise to important ways of looking at measures, distributions, and Hilbert space. The use of the dual space in some fashion is thus characteristic of functional analysis. It is also inherent in the Fourier transform. ...more on Wikipedia about "Dual space"

In mathematics, an eigenfunction of a linear operator A defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has ...more on Wikipedia about "Eigenfunction"

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × VR so that ...more on Wikipedia about "F-space"

In mathematics, a number of fixed point theorems in infinite-dimensional spaces generalise the Brouwer fixed point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations. ...more on Wikipedia about "Fixed point theorems in infinite-dimensional spaces"

In functional analysis and related areas of mathematics an FK-AK space or FK-space with the AK property is an FK-space which contains the space of finite sequences and has a Schauder basis. ...more on Wikipedia about "FK-AK space"

In functional analysis and related areas of mathematics a FK-space or Fréchet coordinate space is a sequence space equipped with a topological structure such that it becomes a Fréchet space. FK-spaces with a normable topology are called BK-spaces. ...more on Wikipedia about "FK-space"

In functional analysis and related areas of mathematics, Fréchet spaces or Frechet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces, normed vector spaces which are complete with respect to the metric induced by the norm. Fréchet spaces, in contrast, are locally convex spaces which are complete with respect to a translation invariant metric. ...more on Wikipedia about "Fréchet space"

Free group factors isomorphism problem is an unsolved mathematical question. Given a free group on some number of generators, we can consider the von Neumann algebra generated by the group algebra. These are important examples of factors. The isomorphism problem asks if these are isomorphic for different numbers of generators. ...more on Wikipedia about "Free group factors isomorphism problem"

Free probability is a mathematical theory which studies non-commutative random variables. The "freeness" property is the analogue of the classical notion of independence, and it is connected with free products. This theory was initiated by Dan Voiculescu around 1986 in order to ...more on Wikipedia about "Free probability" Fast http://www.shortopedia.com functional_analysis

In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematician K. Friedrichs. This extension is particularly useful in situations where an operator may fail to be essentially self-adjoint or whose essential self-adjointness is difficult to show. ...more on Wikipedia about "Friedrichs extension"

In mathematics, Fuglede's theorem is a result in functional analysis. The following version extends the original theorem. ...more on Wikipedia about "Fuglede's theorem"

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. It has its historical roots in the study of transformations, such as the Fourier transform, and in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach. ...more on Wikipedia about "Functional analysis"

In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. If f is a function, say a numerical function of a real number, and M is an operator, there is no particular reason why the expression ...more on Wikipedia about "Functional calculus"