Editing Vector bundle (section)

== Operations on vector bundles ==
Most [[Operation (mathematics)|operations]] on vector spaces can be extended to vector bundles by performing the vector space operation ''fiberwise''.

For example, if ''E'' is a vector bundle over ''X'', then there is a bundle ''E*'' over ''X'', called the '''[[dual bundle]]''', whose fiber at ''x'' ∈ ''X'' is the [[dual vector space]] (''E<sub>x</sub>'')*. Formally ''E*'' can be defined as the set of pairs (''x'', φ), where ''x'' ∈ ''X'' and φ ∈ (''E''<sub>''x''</sub>)*. The dual bundle is locally trivial because the [[transpose|dual space]] of the inverse of a local trivialization of ''E'' is a local trivialization of ''E*'': the key point here is that the operation of taking the dual vector space is [[functorial]].

There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundles ''E'', ''F'' on ''X'' (over the given field). A few examples follow.

* The '''Whitney sum''' (named for [[Hassler Whitney]]) or '''direct sum bundle''' of ''E'' and ''F'' is a vector bundle ''E'' ⊕ ''F'' over ''X'' whose fiber over ''x'' is the [[direct sum of modules|direct sum]] ''E<sub>x</sub>'' ⊕ ''F<sub>x</sub>'' of the vector spaces ''E<sub>x</sub>'' and ''F<sub>x</sub>''.
* The '''[[tensor product bundle]]''' ''E'' ⊗ ''F'' is defined in a similar way, using fiberwise [[tensor product]] of vector spaces.
* The '''Hom-bundle''' Hom(''E'', ''F'') is a vector bundle whose fiber at ''x'' is the space of linear maps from ''E<sub>x</sub>'' to ''F<sub>x</sub>'' (which is often denoted Hom(''E''<sub>''x''</sub>, ''F<sub>x</sub>'') or ''L''(''E''<sub>''x''</sub>, ''F''<sub>''x''</sub>)). The Hom-bundle is so-called (and useful) because there is a [[bijection]] between vector bundle homomorphisms from ''E'' to ''F'' over ''X'' and sections of Hom(''E'', ''F'') over ''X''.
* Building on the previous example, given a section ''s'' of an [[endomorphism]] bundle Hom(''E'', ''E'') and a function ''f'': ''X'' → '''R''', one can construct an '''eigenbundle''' by taking the fiber over a point ''x'' ∈ ''X'' to be the ''f''(''x'')-[[Eigenvector#Eigenspace and spectrum|eigenspace]] of the linear map ''s''(''x''): ''E''<sub>''x''</sub> → ''E''<sub>''x''</sub>.  Though this construction is natural, unless care is taken, the resulting object will not have local trivializations.  Consider the case of ''s'' being the zero section and ''f'' having isolated zeroes.  The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in ''E'', while everywhere else the fiber is the trivial 0-dimensional vector space.
* The [[dual bundle|dual vector bundle]] ''E*'' is the Hom bundle Hom(''E'', '''R''' × ''X'') of bundle homomorphisms of ''E'' and the trivial bundle '''R''' × ''X''.  There is a canonical vector bundle isomorphism Hom(''E'', ''F'') = ''E*'' ⊗ ''F''.

Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the [[category of vector spaces]] can also be performed on the category of vector bundles in a [[functor]]ial manner.  This is made precise in the language of [[smooth functor]]s.  An operation of a different nature is the '''[[pullback bundle]]''' construction. Given a vector bundle ''E'' → ''Y'' and a continuous map ''f'': ''X'' → ''Y'' one can "pull back" ''E'' to a vector bundle ''f*E'' over ''X''. The fiber over a point ''x'' ∈ ''X'' is essentially just the fiber over ''f''(''x'') ∈ ''Y''. Hence, Whitney summing ''E'' ⊕ ''F'' can be defined as the pullback bundle of the diagonal map from ''X'' to ''X'' × ''X'' where the bundle over ''X'' × ''X'' is ''E''&nbsp;×&nbsp;''F''.

'''Remark''': Let ''X'' be a [[compact space]]. Any vector bundle ''E'' over ''X'' is a direct summand of a trivial bundle; i.e., there exists a bundle ''E''{{'}} such that ''E'' ⊕ ''E''{{'}} is trivial. This fails if ''X'' is not compact: for example, the [[tautological line bundle]] over the infinite real projective space does not have this property.{{sfn|Hatcher|2003|loc=Example 3.6}}