infinite-dimensional

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, the dimension of a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''V'' is the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(i.e. the number of vectors) of a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other ba ...
of ''V'' over its base
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
. p. 44, §2.36 It is sometimes called Hamel dimension (after
Georg Hamel Georg Karl Wilhelm Hamel (12 September 1877 – 4 October 1954) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ...

) or algebraic dimension to distinguish it from other types of
dimension In and , the dimension of a (or object) is informally defined as the minimum number of needed to specify any within it. Thus a has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point ...

. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say $V$ is if the dimension of $V$ is
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
, and if its dimension is
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

. The dimension of the vector space $V$ over the field $F$ can be written as $\dim_F\left(V\right)$ or as read "dimension of $V$ over $F$". When $F$ can be inferred from context, $\dim\left(V\right)$ is typically written.

# Examples

The vector space $\R^3$ has $\left\$ as a standard basis, and therefore $\dim_\left(\R^3\right) = 3.$ More generally, $\dim_\left(\R^n\right) = n,$ and even more generally, $\dim_\left(F^n\right) = n$ for any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
$F.$ The complex numbers $\Complex$ are both a real and complex vector space; we have $\dim_\left(\Complex\right) = 2$ and $\dim_\left(\Complex\right) = 1.$ So the dimension depends on the base field. The only vector space with dimension $0$ is $\,$ the vector space consisting only of its zero element.

# Properties

If $W$ is a linear subspace of $V$ then $\dim \left(W\right) \leq \dim \left(V\right).$ To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if $V$ is a finite-dimensional vector space and $W$ is a linear subspace of $V$ with $\dim \left(W\right) = \dim \left(V\right),$ then $W = V.$ The space $\R^n$ has the standard basis $\left\,$ where $e_i$ is the $i$-th column of the corresponding identity matrix. Therefore, $\R^n$ has dimension $n.$ Any two finite dimensional vector spaces over $F$ with the same dimension are isomorphic. Any bijective map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If $B$ is some set, a vector space with dimension $, B,$ over $F$ can be constructed as follows: take the set $F^\left(B\right)$ of all functions $f : B \to F$ such that $f\left(b\right) = 0$ for all but finitely many $b$ in $B.$ These functions can be added and multiplied with elements of $F$ to obtain the desired $F$-vector space. An important result about dimensions is given by the rank–nullity theorem for linear maps. If $F / K$ is a field extension, then $F$ is in particular a vector space over $K.$ Furthermore, every $F$-vector space $V$ is also a $K$-vector space. The dimensions are related by the formula $\dim_K(V) = \dim_K(F) \dim_F(V).$ In particular, every complex vector space of dimension $n$ is a real vector space of dimension $2n.$ Some formulae relate the dimension of a vector space with the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the base field and the cardinality of the space itself. If $V$ is a vector space over a field $F$ then and if the dimension of $V$ is denoted by $\dim V,$ then: :If dim $V$ is finite then $, V, = , F, ^.$ :If dim $V$ is infinite then $, V, = \max \left(, F, , \dim V\right).$

# Generalizations

A vector space can be seen as a particular case of a matroid, and in the latter there is a well-defined notion of dimension. The length of a module and the rank of an abelian group both have several properties similar to the dimension of vector spaces. The Krull dimension of a commutative Ring (algebra), ring, named after Wolfgang Krull (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

## Trace

The dimension of a vector space may alternatively be characterized as the Trace (linear algebra), trace of the identity operator. For instance, $\operatorname\ \operatorname_ = \operatorname \left\left(\begin 1 & 0 \\ 0 & 1 \end\right\right) = 1 + 1 = 2.$ This appears to be a circular definition, but it allows useful generalizations. Firstly, it allows for a definition of a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an Algebra over a field, algebra $A$ with maps $\eta : K \to A$ (the inclusion of scalars, called the ''unit'') and a map $\epsilon : A \to K$ (corresponding to trace, called the ''counit''). The composition $\epsilon \circ \eta : K \to K$ is a scalar (being a linear operator on a 1-dimensional space) corresponds to "trace of identity", and gives a notion of dimension for an abstract algebra. In practice, in bialgebras, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension ($\epsilon := \textstyle \operatorname$), so in these cases the normalizing constant corresponds to dimension. Alternatively, it may be possible to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator". These fall under the rubric of "trace class operators" on a Hilbert space, or more generally nuclear operators on a Banach space. A subtler generalization is to consider the trace of a ''family'' of operators as a kind of "twisted" dimension. This occurs significantly in representation theory, where the Character (mathematics), character of a representation is the trace of the representation, hence a scalar-valued function on a Group (mathematics), group $\chi : G \to K,$ whose value on the identity $1 \in G$ is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: $\chi\left(1_G\right) = \operatorname\ I_V = \dim V.$ The other values $\chi\left(g\right)$ of the character can be viewed as "twisted" dimensions, and find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of monstrous moonshine: the j-invariant, $j$-invariant is the graded dimension of an infinite-dimensional graded representation of the monster group, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.

* * * * * , also called Lebesgue covering dimension

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