In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.
Distributions are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.
In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.
In mathematics, especially functional analysis, a self-adjoint element of a C*-algebra is called positive if its spectrum consists of non-negative real numbers. Moreover, an element of a C*-algebra is positive if and only if there is some in such that . A positive element is self-adjoint and thus normal.
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so continuous.
In mathematics, the ba space of an algebra of sets is the Banach space consisting of all bounded and finitely additive signed measures on . The norm is defined as the variation, that is
In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus, which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential
The spectrum of a linear operator that operates on a Banach space consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts:
In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology.
In mathematical analysis, a Banach limit is a continuous linear functional defined on the Banach space of all bounded complex-valued sequences such that for all sequences , in , and complex numbers :
- (linearity);
- if for all , then (positivity);
- , where is the shift operator defined by (shift-invariance);
- if is a convergent sequence, then .
An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener. Wiener's original construction only applied to the space of real-valued continuous paths on the unit interval, known as classical Wiener space; Leonard Gross provided the generalization to the case of a general separable Banach space.
In functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.
In functional analysis, the dual norm is a measure of the "size" of each continuous linear functional defined on a normed vector space.
In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the weak integral in contrast to the Bochner integral, which is the strong integral.
In mathematics — specifically, in probability theory — the concentration dimension of a Banach space-valued random variable is a numerical measure of how “spread out” the random variable is compared to the norm on the space.
