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In mathematics, a **normed algebra***A* is an algebra over a field which has a sub-multiplicative norm:

In mathematics, an **algebra over a field** is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure, which consists of a set, together with operations of multiplication, addition, and scalar multiplication by elements of the underlying field, and satisfies the axioms implied by "vector space" and "bilinear".

In linear algebra, functional analysis, and related areas of mathematics, a **norm** is a function that assigns a strictly positive *length* or *size* to each vector in a vector space—except for the zero vector, which is assigned a length of zero. A **seminorm**, on the other hand, is allowed to assign zero length to some non-zero vectors.

Some authors require it to have a multiplicative identity 1_{A} such that ║1_{A}║ = 1.

- Banach algebra
- Composition algebra
- Division algebra
- Gelfand–Mazur theorem
- Hurwitz's theorem (composition algebras)

In mathematics, especially functional analysis, a **Banach algebra**, named after Stefan Banach, is an associative algebra *A* over the real or complex numbers that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm. The norm is required to satisfy

In mathematics, a **composition algebra**A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies

In the field of mathematics called abstract algebra, a **division algebra** is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.

"Normed Algebra". *Encyclopaedia of Mathematics*. Retrieved 20 May 2018.

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In abstract algebra, an **alternative algebra** is an algebra in which multiplication need not be associative, only alternative. That is, one must have

In mathematics, more specifically in functional analysis, a **Banach space** 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 vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, a **field** is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do.
A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.

**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 and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

A **vector space** is a collection of objects called **vectors**, which may be added together and multiplied ("scaled") by numbers, called *scalars*. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called *axioms*, listed below.

In mathematics, a **ring** is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

In mathematics, the **Cayley–Dickson construction**, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as **Cayley–Dickson algebras**, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics.

In functional analysis, a discipline within mathematics, given a C*-algebra *A*, the **Gelfand–Naimark–Segal construction** establishes a correspondence between cyclic *-representations of *A* and certain linear functionals on *A*. The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.

In algebra, the **Brahmagupta–Fibonacci identity** expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says

In mathematics, the **Gelfand representation** in functional analysis has two related meanings:

In mathematics, **operator theory** is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis.

In mathematics, a **quaternion algebra** over a field *F* is a central simple algebra *A* over *F* that has dimension 4 over *F*. Every quaternion algebra becomes the matrix algebra by *extending scalars*, i.e. for a suitable field extension *K* of *F*,
is isomorphic to the 2×2 matrix algebra over *K*.

In algebraic number theory, the **prime ideal theorem** is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field *K*, with norm at most *X*.

In mathematics and abstract algebra, a **relation algebra** is a residuated Boolean algebra expanded with an involution called **converse**, a unary operation. The motivating example of a relation algebra is the algebra 2^{X²} of all binary relations on a set *X*, that is, subsets of the cartesian square *X*^{2}, with *R*•*S* interpreted as the usual composition of binary relations *R* and *S*, and with the converse of *R* as the converse relation.

In mathematics, the **Banach–Stone theorem** is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.

In mathematics, **Hurwitz's theorem** is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called **Hurwitz algebras**, are examples of composition algebras.

In mathematics, an **algebraic number field** *F* is a finite degree field extension of the field of rational numbers **Q**. Thus *F* is a field that contains **Q** and has finite dimension when considered as a vector space over **Q**.

In mathematics, the **Hurwitz problem**, named after Adolf Hurwitz, is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables.