Functional analysis

property of binary operations, for which changing the order of the operands does not change the result

property of binary operations allowing sequences of operations to be regrouped without changing their value

inner product space that is metrically complete; a Banach space whose norm induces an inner product (follows the parallelogram identity)

branch of mathematical analysis concerned with infinite-dimensional topological vector spaces, often spaces of functions

function between ordered sets that preserves or reverses the given order

thumb|400px|class=skin-invert-image|Visual comparison of convolution, cross-correlation, and [[autocorrelation. For the operations involving function f, and assuming the height of f is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point. The symmetry of f is the reason f \star g and g*f are identical in this example. ]]

length in a vector space

normed vector space that is complete

method for orthonormalising a set of vectors

mapping from one vector space or module to another in mathematics

vector space of linear functionals (may consist only on continuous functionals or of all functionals)

set is called bounded, if it is, in a certain sense, of finite size

inverse image of zero under a linear map

subset of a vector space that forms a vector space itself

matrix describing a quantum system in a pure or mixed state, a statistical mixture of quantum states

a continuous functional on a space of test functions (Schwartz space), which generalizes the concept of locally integrable functions

mathematical methods used to find an approximate solution to a problem which cannot be solved exactly

matrix decomposition

theorem in topology and functional analysis

linear mapping from a vector space into its field of scalars

linear transformation that, when applied multiple times to any value, gives the same result as if it were applied once

basis of a normed space consisting of mutually orthogonal elements of norm 1

criterion about convergence of series

theorem that a pointwise bounded set of linear operators on a Banach space is uniformly bounded in operator norm

right|frame|This solution of the vibrations of a circular drum|vibrating drum problem is, at any point in time, an eigenfunction of the [[Laplace operator on a disk.]]

construct related to weighted sums and averages

term used in functional analysis

set of points in 3D space of distance 1 from a fixed central point

In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:

concept in linear algebra

vector space consisting of affine subsets

property of point sets in Euclidean spaces

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the two vectors are perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.

map that assigns a length or size to any operator in a function space

mathematical technique used in data compression and analysis

right|thumb|A solution to a discretized partial differential equation, obtained with the finite element method.

set of a ring's prime ideals

weak derivation

type of continuity of a complex-valued function

topological space in which every intersection of countably many dense open subsets is dense

type of function

map taking two vectors from a complex vector space and returning a complex number, which is linear in one variable and semilinear in another variable

topological vector space in which every vector has a convex neighborhood

function whose squared absolute value has finite integral

function made from a set

uniform space which has finite covers consisting of "arbitrarily small" sets

Computational tool

lemma

branch of functional analysis

mathematical series

mathematical concept

operator on functions, defined by the composition of Fourier transformation, multiplication with a certain smooth function of both position and momentum, and inverse Fourier transformation

p-norm for p equal to ∞

Topologies on the set of operators on a Hilbert space

thumb|A mollifier (top) in Dimension (mathematics)|dimension one. At the bottom, in red is a function with a corner (left) and sharp jump (right), and in blue is its mollified version.

Mathematical term

Hyperplane in geometry

Type of function in linear algebra

transformation preserving the inner product