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A [[Fibre space|fibre space]]  $  \pi :  X \rightarrow B $
each fibre  $  \pi  ^ {-1} ( b) $
of which is endowed with the structure of a (finite-dimensional) [[Vector space|vector space]]  $  V $
over a skew-field  $  {\mathcal P} $
such that the following local triviality condition is satisfied. Each point  $  b \in B $
has an open neighbourhood  $  U $
and an  $  U $-isomorphism of fibre bundles  $  \phi :  \pi  ^ {-1} ( U) \rightarrow U \times V $
such that  $  \phi \mid  _ {\pi  ^ {- 1} ( b) } :  \pi  ^ {-1} ( b) \rightarrow b \times V $
is an isomorphism of vector spaces for each  $  b \in B $; 
$   \mathop{\rm dim}  V $
is said to be the dimension of the vector bundle. The sections of a vector bundle  $  \pi $
form a locally free module  $  \Gamma ( \pi ) $
over the ring of continuous functions on  $  B $
with values in  $  {\mathcal P} $. 
A morphism of vector bundles is a morphism of fibre bundles  $  f:  \pi \rightarrow \pi  ^  \prime  $
for which the restriction to each fibre is linear. The set of vector bundles and their morphisms forms the category  $  \mathbf{Bund} $. 
The concept of a vector bundle arose as an extension of the [[Tangent bundle|tangent bundle]] and the [[Normal bundle|normal bundle]] in differential geometry; by now it has become a basic tool for studies in various branches of mathematics: differential and algebraic topology, the theory of linear connections, algebraic geometry, the theory of (pseudo-) differential operators, etc.

A subset  $  X  ^  \prime  \subset  X $
such that  $  \pi \mid  _ {X}  ^  \prime  :  X  ^  \prime  \rightarrow B $
is a vector bundle and  $  X  ^  \prime  \cap \pi  ^ {-1} ( b) $
is a vector subspace in  $  \pi  ^ {-1} ( b) $
is said to be a subbundle of the vector bundle  $  \pi $. 
For instance, let  $  V $
be a vector space and let  $  G _ {k} ( V) $
be the [[Grassmann manifold|Grassmann manifold]] of subspaces of  $  V $
of dimension  $  k $; 
the subspace of the product  $  G _ {k} ( V) \times V $, 
consisting of pairs  $  ( p, v) $
such that  $  v \in p $, 
will then be a subbundle  $  \gamma _ {k} $
of the trivial vector bundle  $  G _ {k} ( V) \times V $. 
The union of all vector spaces  $  \pi  ^ {-1} ( b) / \pi _ {2}  ^ {-1} ( b) $, 
where  $  \pi _ {2} $
is a subbundle of  $  \pi $
endowed with the quotient topology, is said to be a quotient bundle of  $  \pi $. 
Let, furthermore,  $  V $
be a vector space and let  $  G  ^ {k} ( V) $
be the Grassmann manifold of subspaces of  $  V $
of codimension  $  k $; 
the quotient bundle  $  \gamma  ^ {k} $
of the trivial vector bundle  $  G  ^ {k} ( V) \times V $
is defined as the quotient space of the product  $  G  ^ {k} ( V) \times V $
by the subbundle consisting of all pairs  $  ( p, v) $, 
$  v \in p $. 
The concepts of a subbundle and a quotient bundle are used in contraction and glueing operations used to construct vector bundles over quotient spaces.

A  $  B $-
morphism of vector bundles  $  f :  \pi \rightarrow \pi  ^  \prime  $
is said to be of constant rank (pure) if  $   \mathop{\rm dim}   \mathop{\rm ker}  f \ \mid  _ {\pi  ^ {- 1} ( b) } $
is locally constant on  $  B $. 
Injective and surjective morphisms are exact and are said to be monomorphisms and epimorphisms of the vector bundle, respectively. The following vector bundles are uniquely defined for a morphism  $  f $
of locally constant rank:  $   \mathop{\rm Ker}  f $ (the kernel of  $  f  $), 
which is a subbundle of  $  \pi $; 
$   \mathop{\rm Im}  f $ (the image of  $  f  $), 
which is a subbundle of  $  \pi  ^  \prime  $; 
$   \mathop{\rm Coker}  f $ (the cokernel of  $  f  $), 
which is a quotient bundle of  $  \pi $; 
and  $   \mathop{\rm Coim}  f $ (the co-image of  $  f  $), 
which is a quotient bundle of  $  \pi  ^  \prime  $. 
Any subbundle  $  \pi _ {1} $
is the image of some monomorphism  $  i:  \pi _ {1} \rightarrow \pi $, 
while any quotient bundle  $  \pi _ {2} $
is the cokernel of some epimorphism  $  j :  \pi \rightarrow \pi _ {2} $. 
A sequence of  $  B $-
morphisms of vector bundles

$$ 
{} \dots \rightarrow  \pi  ^  \prime   \rightarrow  \pi  \rightarrow  \pi  ^ {\prime\prime}  \rightarrow \dots
$$

is said to be exact if the sequence

$$ 
{} \dots \rightarrow  ( \pi  ^  \prime  )  ^ {-1} ( b)  \rightarrow  \pi  ^ {-1} ( b)
 \rightarrow  ( \pi  ^ {\prime\prime} )  ^ {-1} ( b)  \rightarrow \dots
$$

is exact for all  $  b \in B $. 
In particular, the sequence

$$ 
0  \rightarrow  \pi _ {1}  \rightarrow ^ { i }   \pi  \rightarrow ^ { j }   \pi _ {2}  \rightarrow  0 ,
$$

where  $  0 $
is the zero vector bundle, is exact if  $  i $
is a monomorphism,  $  j $
is an epimorphism and  $   \mathop{\rm Im}  i = { \mathop{\rm Ker} }  j $. 
The set of vector bundles over  $  B $
and  $  B $-
morphisms of locally constant rank forms an exact subcategory  $  \mathbf{Bund} _ {B} $
of the category  $  \mathbf{Bund} $.

For any vector bundle  $  \pi :  X \rightarrow B $
and mapping  $  u:  B _ {1} \rightarrow B $, 
the [[Induced fibre bundle|induced fibre bundle]]  $  u  ^ {*} ( \pi ) $
is endowed with a vector bundle structure such that the morphism  $  U:  u  ^ {*} ( \pi ) \rightarrow \pi $
is a vector bundle morphism. This structure is unique and has the following property: Every fibre mapping  $  {( u  ^ {*} ( \pi )) }  ^ {-1} ( b) \rightarrow \pi  ^ {-1} ( u( b)) $
is an isomorphism of vector spaces. For instance, a vector bundle of dimension  $  k $
over a paracompact space  $  B $
is isomorphic to one of the vector bundles  $  u  ^ {*} ( \gamma _ {k} ) $
and  $  \widetilde{u}  {}  ^ {*} ( \gamma  ^ {k} ) $
induced by certain mappings  $  u:  B \rightarrow G _ {k} ( V) $
and  $  \widetilde{u}  :  B \rightarrow G  ^ {k} ( V) $, 
respectively; moreover, homotopic mappings induce isomorphic vector bundles and, if  $   \mathop{\rm dim}  V \neq \infty $, 
the converse is true: To isomorphic vector bundles there correspond homotopic mappings  $  u $
and  $  \widetilde{u}  $. 
This is one of the fundamental theorems in the homotopic classification of vector bundles, expressing the universal character of the vector bundles  $  \gamma _ {k} $
and  $  \gamma  ^ {k} $
with respect to the classifying mappings  $  u $
and  $  \widetilde{u}  $.

Any continuous operation ([[Functor|functor]])  $  T $
on the category of vector spaces uniquely determines a continuous functor on the category of vector bundles over  $  B $; 
in this way it is possible to construct bundles associated with a given vector bundle: tensor bundles, vector bundles of morphisms  $  { \mathop{\rm Hom} } _ {B} ( \pi , \pi  ^  \prime  ) $
and, in particular, the dual vector bundle  $  \pi  ^ {*} $, 
exterior powers of vector bundles, etc., whose sections are vector bundles with supplementary structures. These are extensively employed in practical applications.

A direct sum (Whitney sum)  $  \pi \oplus \pi  ^  \prime  $
and tensor product  $  \pi \otimes \pi  ^  \prime  $
have been defined for two vector bundles  $  \pi $
and  $  \pi  ^  \prime  $. 
With respect to these operations the set of classes  $  { \mathop{\rm Vekt} } _ {B} $
of isomorphic vector bundles over  $  B $
forms a semi-ring which plays an important part in the construction of a [[K-functor| $  K $-
functor]]; thus, if for vector bundles  $  \pi $
and  $  \pi  ^  \prime  $
there exist trivial vector bundles  $  \theta $
and  $  \theta  ^  \prime  $
such that the vector bundles  $  \pi \oplus \theta $
and  $  \pi  ^  \prime  \oplus \theta  ^  \prime  $
are isomorphic (i.e.  $  \pi $
and  $  \pi  ^  \prime  $
are stably equivalent), then their images in the "completion"  $  K ( B) $
of the semi-ring  $  { \mathop{\rm Vekt} } _ {B} $
are identical; moreover, the fact that the ring  $  K( B) $
and the set of classes of stably-equivalent vector bundles coincide follows from the existence of an inverse vector bundle for any vector bundle over a paracompact space.

For any vector bundle  $  \pi :  X \rightarrow B $
over a paracompact space  $  B $
there exists a section  $  \beta $
of the vector bundle

$$ 
\pi  ^ {*} \oplus \pi  ^ {*}  =   \mathop{\rm Hom} ( \pi \oplus \pi , P ),
$$

where  $  P $
is a trivial one-dimensional vector bundle, which on each fibre  $  \pi  ^ {-1} ( b) $
is a positive-definite form, i.e.  $  \pi $
is metrizable; this makes it possible to establish, in particular, the splittability of an arbitrary exact sequence of vector bundles

$$ 
0  \rightarrow  \xi  \rightarrow ^ { u }   \pi  \rightarrow ^ { v }   \zeta  \rightarrow  0
$$

in which  $  \pi $
is metrizable, that is, the existence of a morphism  $  w :  \xi \oplus \zeta \rightarrow \pi $
such that  $  wi = u $, 
$  vw = j $, 
where  $  i $
is the imbedding into the first term and  $  j $
is the projection onto the second term.

If, in each fibre  $  \pi  ^ {-1} ( b) $
of the vector bundle  $  \pi :  X \rightarrow B $, 
one identifies the points lying on the same line passing through zero, one obtains a bundle  $  \pi _ {0} :  \Pi _ {( \pi ) }  \rightarrow B $, 
which is associated with  $  \pi $
and is said to be its projectivization; a fibre of  $  \pi _ {0} $
is the projective space  $  \Pi ( V) $
which is associated with  $  V $. 
This bundle is used to study Thom spaces  $  T ( \pi ) = \Pi ( \pi \oplus P)/ \Pi ( \pi ) $ (cf. [[Thom space|Thom space]]), used in the homotopic interpretation of classes of bordant manifolds, characteristic classes of vector bundles describing the homological properties of manifolds, etc.

The concept of a vector bundle can be generalized to the case when the fibre is an infinite-dimensional vector space; in doing so, one must distinguish between the different topologies of the space of morphisms  $  { \mathop{\rm Hom} } ( \pi , \pi  ^  \prime  ) $, 
suitably modify the definitions of a pure morphism and an exact sequence of morphisms, and also the construction of vector bundles associated with continuous functors on the category of infinite-dimensional vector spaces.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) {{MR|0242081}} {{ZBL|0653.53001}} {{ZBL|0284.53018}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.F. Atiyah, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380142.png" />-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III {{MR|1931083}} {{MR|1532744}} {{MR|0155257}} {{ZBL|1008.57001}} {{ZBL|0103.15101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) {{MR|0533884}} {{ZBL|0444.32004}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) {{MR|1335917}} {{MR|0202713}} {{ZBL|0376.14001}} </TD></TR></table>

====Comments====
For more on the universality and classifying properties of the bundles  $  \gamma  ^ {k} $
and  $  \gamma _ {k} $
cf. [[Classifying space|Classifying space]] or [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) {{MR|0440554}} {{ZBL|0298.57008}} </TD></TR></table>