This construction can be carried out more generally: for a commutative ringR one can define the dual numbers over R as the quotient of the polynomial ringR[X] by the ideal (X2): the image of X then has square equal to zero and corresponds to the element ε from above. This ring and its generalisations play an important part in the algebraic theory of derivations and Kähler differentials (purely algebraic differential forms).
Over any ring R, the dual number a + bε is a unit (i.e. multiplicatively invertible) if and only if a is a unit in R. In this case, the inverse of a + bε is a-1 - ba-2ε. As a consequnce, we see that the dual numbers over any field (or any commutative local ring) form a local ring.
In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself.
The continuous dual V′ of a normed vector space V (e.g., a Banach space or a Hilbert space) forms a normed vector space.
This turns the continuous dual into a normed vector space, indeed into a Banach space so long as the underlying field is complete which is often included in the definition of the normed vector space.
Feeds |
Contact