12 Facts About Vector space

An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms say that a vector space is an abelian group under addition, and the four remaining axioms say that the scalar multiplication defines a ring homomorphism from the field into the endomorphism ring of this group.

Vector space envisaged sets of abstract objects endowed with operations.

Elements of form a vector space that is usually denoted and called a coordinate space.

The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic.

Via the injective natural map, any vector space can be embedded into its bidual; the map is an isomorphism if and only if the space is finite-dimensional.

The set of all eigenvectors corresponding to a particular eigenvalue of forms a vector space known as the eigenspace corresponding to the eigenvalue in question.

Tensor product is a particular vector space that is a universal recipient of bilinear maps g, as follows.

Denotes the limit of the corresponding finite partial sums of the sequence i?N of elements of V For example, the fi could be functions belonging to some function space V, in which case the series is a function series.

Roughly, a vector space is complete provided that it contains all necessary limits.

Vector bundles over X are required to be locally a product of X and some vector space V: for every x in X, there is a neighborhood U of x such that the restriction of p to p is isomorphic to the trivial bundle.

Nevertheless, a vector space can be compactly defined as a module over a ring which is a field, with the elements being called vectors.

In particular, a vector space is an affine space over itself, by the map.