To illustrate, take again the finite-dimensional case.

Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as

finite-dimensional case, that is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued multiplication operators.

Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection J from the definition is bijective, by the rank-nullity theorem.

This is not the case in a finite-dimensional Hilbert space, where every operator is bounded-it has a largest eigenvalue.

The antipode S is sometimes required to have a K-linear inverse, which is automatic in the finite-dimensional case, or if H is commutative or cocommutative ( or more generally quasitriangular ).

In the finite-dimensional case, this formulation is susceptible to a generalization: if

In the specific case where G is a finite-dimensional Lie algebra ( as a special case of a Kac-Moody algebra ), then the irreducible representations with dominant integral highest weights are also finite-dimensional.

In the case of operators on finite-dimensional vector spaces, for complex square matrices, the relation of being isospectral for two diagonalizable matrices is just similarity.

In the group case it says that if M and N are two finite-dimensional irreducible representations

In the finite-dimensional case, a matrix can always be decomposed in the form UDW, where U and W are unitary matrices and D is a diagonal matrix with the singular values lying on the diagonal.

In the general setting, then, an automorphic form is a function F on G ( with values in some fixed finite-dimensional vector space V, in the vector-valued case ), subject to three kinds of conditions:

In the vector-valued case the specification can involve a finite-dimensional group representation ρ acting on the components to ' twist ' them.

In the case H is finite-dimensional, that is,, the set of projective rays may be treated just as any other projective space ; it is a homogeneous space for a unitary group or orthogonal group, in the complex and real cases respectively.

Even the case of representations of the polynomial algebra in a single variable are of interest – this is denoted by and is used in understanding the structure of a single linear operator on a finite-dimensional vector space.

In that case, there are finite-dimensional modules for the group algebra that are not projective modules.

Let V and W be finite-dimensional real or complex vector spaces, with n = dim W. Let A < sub > 1 </ sub >, ..., A < sub > n − 1 </ sub > be analytic functions with values in End ( V ) and b an analytic function with values in V, defined on some neighbourhood of ( 0, 0 ) in V × W. In this case, the same result holds.

For the case of a finite-dimensional graph ( having a finite number of edges and vertices ), the discrete Laplace operator is more commonly called the Laplacian matrix.

( For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.

This is a special case of the equivalence of norms in finite-dimensional Normed vector spaces.

Hilbert himself proved the finite generation of invariant rings in the case of the field of complex numbers for some classical semi-simple Lie groups ( in particular the general linear group over the complex numbers ) and specific linear actions on polynomial rings, i. e. actions coming from finite-dimensional representations of the Lie-group.

This is not true for infinite dimensions ; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.

In mathematics, any vector space, V, has a corresponding dual vector space ( or just dual space for short ) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors.

More generally any quadratically closed subfield of or will suffice for this purpose, e. g., the algebraic numbers, but when it is a proper subfield ( i. e., neither nor ) even finite-dimensional inner product spaces will fail to be metrically complete.

By transferring upper triangularisation of operators of finite-dimensional complex vector space, there is an internal orthonormal Hilbert space basis for where runs from to, such that each of the corresponding-dimensional subspaces is-invariant.

Every finite-dimensional vector space is isomorphic to its dual space, but this isomorphism relies on an arbitrary choice of isomorphism ( for example, via choosing a basis and then taking the isomorphism sending this basis to the corresponding dual basis ).

On the category of finite-dimensional vector spaces and linear maps, one can define an infranatural isomorphism from vector spaces to their dual by choosing an isomorphism for each space ( say, by choosing a basis for every vector space and taking the corresponding isomorphism ), but this will not define a natural transformation.

In mathematics, the L < sup > p </ sup > spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.

The dynamics for the rigid body take place in a finite-dimensional configuration space ; the dynamics for the fluid occur in an infinite-dimensional configuration space.

This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces.

Furthermore, given such a group G, Langlands constructs the Langlands dual group < sup > L </ sup > G, and then, for every automorphic cuspidal representation of G and every finite-dimensional representation of < sup > L </ sup > G, he defines an L-function.

If X is a Hilbert space and T is a normal operator, then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators ( Hermitian matrices, for example ).

Wedderburn's structure theorems were formulated for finite-dimensional algebras over a field while Artin generalized them to Artinian rings.

# for every x in X, the structure of a finite-dimensional real vector space on the fiber π < sup >− 1 </ sup >(

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Nonetheless, whether we are working in an infinite-dimensional or finite-dimensional Hilbert space, the role of an observable in quantum mechanics is to assign real numbers to outcomes of particular measurements ; this means that only certain measurements can determine the value of an observable for some state of a quantum system.

In particular, for any field K, any finite-dimensional module for a finite-dimensional algebra over K has a composition series, unique up to equivalence.

* Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional.

A real Lie group is a group which is also a finite-dimensional real smooth manifold, and in which the group operations of multiplication and inversion are smooth maps.

In mathematics, specifically in real analysis, the Bolzano – Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R < sup > n </ sup >.

More generally we consider a Hermitian map A on a finite-dimensional real or complex inner product space V endowed with a positive definite Hermitian inner product.

The finite-dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix.

* Any open subset of a finite-dimensional real, and therefore complex, vector space is a diffeological space.

By the finite-dimensional spectral theorem, such operators can be associated with an orthonormal basis of the underlying space in which the operator is represented as a diagonal matrix with entries in the real numbers.

If a normal operator on a finite-dimensional real or complex Hilbert space ( inner product space ) stabilizes a subspace, then it also stabilizes its orthogonal complement.

Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E < sub > 6 </ sub > coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all adjoint forms of E < sub > 6 </ sub > have fundamental group Z / 3Z in the sense of algebraic geometry, with Galois action as on the third roots of unity ; this means that they admit exactly one triple cover ( which may be trivial on the real points ); the further non-compact real Lie group forms of E < sub > 6 </ sub > are therefore not algebraic and admit no faithful finite-dimensional representations.

A partial converse to this statement says that every representation of a finite-dimensional ( real or complex ) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.

* Consider the category of finite-dimensional real vector spaces, and the category of all real matrices ( the latter category is explained in the article on additive categories ).

In mathematics, a generalized flag variety ( or simply flag variety ) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold.

Note that if R is a finite-dimensional simple algebra over a division ring E, D need not be contained in E. For example, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers.

The finite-dimensional division algebras with center R ( that means the dimension over R is finite ) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – M ( n, R ) or M ( n, H ) – is a CSA over the reals, but not a division algebra ( if ).