Dual vector
In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars. In ℝn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the matrix product with the row vector on the left and the column vector on the right. In general, if V is a vector space over a field k, then a linear functional f is a function from V to k that is linear:
The set of all linear functionals from V to k, Homk(V,k), forms a vector space over k with the addition of the operations of addition and scalar multiplication (defined pointwise). This space is called the dual space of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space. It is often written V∗ or V′ when the field k is understood.
If V is a topological vector space, the space of continuous linear functionals — the continuous dual — is often simply called the dual space. If V is a Banach space, then so is its (continuous) dual. To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual. In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, although this is not true in infinite dimensions.
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