§ 7. Vector bundles

Throughout this paragraph, the letter B denotes a manifold of class $C^r$ ($r \geqslant 1$) and the letter M denotes a set endowed with a mapping $\pi$ from M into B. We say that B is the base of M and for every $b \in B$, we denote by $M_b$ and call the fiber of M at $b$ the subset $\pi^{-1}(b)$ of M.

7.1. Definition of vector bundles

7.1.1. A vector chart of M is a triple $t = (U, \varphi, F)$, where U is an open set of B, where F is a Banach space and $\varphi$ a bijection of $\pi^{-1}(U)$ onto $U \times F$ such that $\pi(\varphi^{-1}(b, h)) = b$ for all $b \in B$ and all $h \in F$. We say that U is the domain of the vector chart t and that t is a vector chart of M at $b \in B$ if $b \in U$. For every $b \in U$, we denote by $t_b$ the bijection of F onto $M_b$ defined by $t_b(h) = \varphi^{-1}(b, h)$ for $h \in F$.

7.1.2. We say that two vector charts $t = (U, \varphi, F)$ and $t' = (U', \varphi', F')$ of M are $C^r$-compatible (or simply compatible) if there exists a mapping $\lambda$ of class $C^r$ of the manifold $U \cap U'$ into the Banach space $\mathscr{L}(F; F')$ such that:

7.1.3. We say that a set of vector charts of M is a $C^r$-vector atlas (or simply vector atlas) of M if it consists of vector charts which are pairwise $C^r$-compatible and whose domains have B as their union. We say that two vector atlases $\mathscr{A}$ and $\mathscr{B}$ of M are $C^r$-equivalent (or equivalent) if $\mathscr{A} \cup \mathscr{B}$ is again a vector atlas of M. This relation is an equivalence relation.

7.1.4. A structure of vector bundle of class $C^r$ (with base B) on M is the data of an equivalence class of vector atlases (Ens., Chap. II, § 6, No. 9). A vector chart belonging to a vector atlas of this class is called a vector chart of the vector bundle M.

Let M be a vector bundle with base B. For every $b \in B$, there exists on the fiber $M_b$ one and only one Banach space structure such that, for every vector chart $t = (U, \varphi, F)$ of the vector bundle M at $b$, the mapping $t_b$ is an isomorphism of F onto $M_b$.

Let $c = (U, \psi, E)$ be a chart of the manifold B and let $t = (U, \varphi, F)$ be a vector chart of the vector bundle M, with the same domain U. For $x \in \pi^{-1}(U)$, put:

Then the triple $(\pi^{-1}(U), \alpha, E \times F)$ is a chart of the set $M$. There exists on $M$ one and only one manifold structure of class $C'$ (said to be underlying $M$) for which all the charts thus obtained are charts of the manifold $M$. The triple $(M, B, \pi)$ is then a fibration (6.1.1).

7.1.5. Let $F$ be a Banach space. Put $M = B \times F$, the mapping being the first projection. There exists on $M$ a vector bundle structure (with basis $B$) and only one for which $(B, \mathrm{Id}_M, F)$ is a vector chart. One says that $B \times F$ endowed with this structure is the trivial vector bundle with basis $B$ and fibre $F$ and one sometimes denotes it by $F_B$. The manifold structure of $F_B$ is the product manifold structure and for every $b \in B$, the mapping $h \mapsto (b, h)$ is an isomorphism of Banach spaces of $F$ onto the fibre of $F_B$ at the point $b \in B$. One often denotes by $0$ a trivial vector bundle whose fibre is reduced to $0$.

7.1.6. Let $M$ be a vector bundle with basis $B$. For $b \in B$, one calls rank of $M$ at $b$ and denotes by $\mathrm{rg}_b(M)$, the dimension (finite or $+\infty$) of the Banach space $M_b$. One has $\dim_x M = \dim_b B + \mathrm{rg}_b M$ for $b = \pi(x)$. The function $b \mapsto \mathrm{rg}_b M$ is locally constant. One says that $M$ is of finite rank if $\mathrm{rg}_b M < +\infty$ for every $b \in B$.

7.2. Morphisms of vector bundles

7.2.1. Let $B$ and $B'$ be two manifolds and let $f$ be a morphism from $B$ into $B'$. Let $M$ be a vector bundle with basis $B$ and $M'$ a vector bundle with basis $B'$. One says that a mapping $g$ from $M$ into $M'$ is an $f$-morphism of vector bundles if the following condition is satisfied:

For every point $b_0 \in B$, there exists a vector chart $t = (U, \varphi, F)$ of $M$ at $b_0$, a vector chart $t' = (U', \varphi', F')$ of $M'$ at $f(b_0)$ and a mapping $\lambda$ of class $C'$ from $U$ into $\mathscr{L}(F; F')$ such that $f(U) \subset U'$ and that $g_b \circ t_b = t'_{f(b)} \circ \lambda(b)$ for every $b \in U$, where $g_b$ is the restriction of $g$ to $M_b$.

Under these hypotheses, $g$ is a morphism of varieties and, for every $b \in B$, $g$ induces a continuous linear mapping of $M_b$ into $M'_{f(b)}$. The vector rank of $g$ at $b \in B$ is called and is denoted by $\mathrm{rg}_b(g)$ the rank (finite or $+\infty$) of the linear mapping $g_b$.

Conversely, if $r \geqslant \infty$ or if $M$ is of finite rank, an $f$-morphism of fibrations of $(M, B, \pi)$ into $(M', B', \pi')$ which induces on each fibre $M_b$ a linear mapping of $M_b$ into $M'_{f(b)}$ (for every $b \in B$), is an $f$-morphism of vector bundles.

7.2.2. Let in addition $f'$ be a morphism of $B'$ into a variety $B''$. If $g$ is an $f$-morphism of $M$ into $M'$, and if $g'$ is an $f'$-morphism of $M'$ into a vector bundle $M''$ with base $B''$, the mapping $g' \circ g$ is an $(f' \circ f)$-morphism of $M$ into $M''$. We have $(g' \circ g)_b = g'_{f(b)} \circ g_b$ for every $b$ in $B$.

7.2.3. Let M and M' be two vector bundles with the same base B. A B-morphism, or simply a morphism of M into M', is called every Id_B-morphism. The composite of two morphisms is a morphism.

Let g be a morphism of vector bundles of M into M'; if g is bijective, it is an isomorphism of the variety M onto the variety M', the reciprocal mapping $g^{-1}$ is a morphism of vector bundles of M' into M and we have $(g^{-1})_b = g_b^{-1}$ for every b in B. The mapping g is then an isomorphism of vector bundles.

7.2.4. Let f be a morphism of a variety B' into B and let M be a vector bundle with base B. Put $M' = B' \times_B M$ and denote by $\pi'$ (resp. g) the restriction to M' of the projection of $B' \times M$ onto B' (resp. M). There exists on M' one and only one structure of vector bundle with base B' (relative to $\pi'$) for which g is a f-morphism. We say that M' is the vector bundle with base B' inverse image of M by f and denote it by $f^*M$; the f-morphism g is called the canonical f-morphism of $f^*M$ into M.

The structure of variety of $f^*M$ is that of the fiber product of the varieties B' and M over B (5.11.2); for every $b \in B'$, the mapping $x \mapsto (b, x)$ is an isomorphism of Banach spaces of $M_{f(b)}$ onto $(f^*M)_b$.

The formation of inverse images of vector bundles is transitive.

Let N' be a vector bundle with base B' and let h be a f-morphism of N' into M. There exists one and only one B'-morphism $\tilde{h}$ of N' into $f^*M$ such that $h = g \circ \tilde{h}$.

Let N be a vector bundle with base B and v a B-morphism of N into M. There exists one and only one B'-morphism, denoted by $f^*v$, of $f^*N$ into $f^*M$ such that the diagram

is commutative.

7.2.5. Let B' be a subvariety of B and let i be the canonical injection of B' into B. If M is a vector bundle with base B, the inverse image $i^*M$ is called the vector bundle induced on B' by M and is denoted by $M|B'$. If $t = (U, \varphi, F)$ is a vector chart of M, then $(U \cap B', \varphi|\pi^{-1}(U \cap B'), F)$ is a vector chart of $M|B'$. The canonical f-morphism of $M|B'$ into M is an isomorphism of varieties of $M|B'$ onto the subvariety $\pi^{-1}(B')$ of M.

7.2.6. Let f be a morphism of a variety B' into B and let M (resp. M') be a vector bundle with base B (resp. B'). A f-comorphism of M into M' is called a B'-morphism of $f^*M$ into M'. When B = B' and $f = \mathrm{Id}_B$, the data of a $f$-comorphism of $M$ into $M'$ is equivalent to the data of a $B$-morphism of $M$ into $M'$.

Let $g$ be a $f$-comorphism of $M$ into $M'$. For every $b \in B'$, the mapping $g_b : x \mapsto g(b, x)$ of $M_{f(b)}$ into $M'_b$ is a continuous linear mapping.

Suppose moreover that $f'$ is a morphism of a variety $B''$ into $B'$ and that $h$ is a $f'$-comorphism of $M'$ into a vector bundle $M''$ with base $B''$. The mapping $h \circ f^*g$ of ${f'}^*(f^*M) = (f \circ f')^*M$ into $M''$ is then a $(f \circ f')$-comorphism of $M$ into $M''$, denoted by $h \circ g$. For $b \in B''$, one has

7.3. Multilinear morphisms

7.3.1. Let $M_1, \ldots, M_d$ and $N$ be vector bundles with base $B$, and let $u$ be a mapping of the set $M_1 \times_B \cdots \times_B M_d$ into $N$. We say that $u$ is a multilinear morphism (or $d$-linear) if the following condition is satisfied:

For every $b_0 \in B$, there exist an open neighbourhood $U$ of $b_0$ in $B$, vector bundle charts $t^j = (U, \varphi^j, F^j)$ of $M_j$ (for $1 \leq j \leq d$) and $t = (U, \varphi, F)$ of $N$, and a mapping $\lambda$ of class $C^r$ from $U$ into the Banach space $\mathscr{L}(F^1, \ldots, F^d; F)$ of continuous $d$-linear mappings from $F^1 \times \cdots \times F^d$ into $F$, such that:

for every $b \in U$ and all $x_j \in F^j$.

Every multilinear morphism $u$ is a morphism of manifolds from the fibre product $M_1 \times_B \cdots \times_B M_d$ into $N$ and induces for every $b \in B$ a continuous $d$-linear mapping $u_b$ from $(M_1)_b \times \cdots \times (M_d)_b$ into $N_b$.

If $f$ is a morphism from $B$ into a manifold $B'$, a multilinear morphism from $M_1 \times_B \cdots \times_B M_d$ into a vector bundle $M'$ with base $B'$ is by definition the composite of a multilinear morphism from $M_1 \times_B \cdots \times_B M_d$ into $f^*M'$ with the canonical $f$-morphism of $f^*M'$ into $M'$.

A bilinear morphism is also called a pairing. For $d = 1$, a linear morphism is a morphism in the sense of 7.2.1. For $d = 0$, a 0-linear morphism is identified with a section of $N$ (7.4).

7.3.2. A bundle of algebras of base $B$ is called a vector bundle $A$ of base $B$, endowed with a pairing of $A \times_B A$ into $A$. Each fibre $A_b$ is then endowed with a structure of $K$-algebra. If, for every $b \in B$, the algebra $A_b$ has a unit element, denoted by $e_b$, the mapping $b \mapsto e_b$ is a section of $A$ (cf. 7.4). A bundle of algebras $A$ is said to be locally trivial if, for every point $b_0$ of $B$, there exists a vector chart $t = (U, \varphi, E)$ of $A$ at the point $b_0$ and a structure of $K$-algebra on $E$ such that $t_b$ is an isomorphism of algebras of $E$ onto $A_b$ for every $b \in U$.

7.3.3. Let $A$ be a bundle of associative algebras with unit element of base $B$. A bundle of A-modules of base $B$ is called a vector bundle $M$ of base $B$ endowed with a pairing $m : A \times_B M \to M$ such that the mapping $m_b : A_b \times M_b \to M_b$ defines, for every $b \in B$, a structure of $A_b$-module on the fibre $M_b$.

Let $M$ and $M'$ be two bundles of $A$-modules. An A-homomorphism of $M$ into $M'$ is called any morphism $g : M \to M'$ of vector bundles of base $B$ which induces for every $b$ in $B$ an $A_b$-linear mapping of $M_b$ into $M'_b$.

Suppose $A$ is locally trivial. A fibré $M$ in $A$-modules is said to be locally trivial if, for every point $b_0$ of $B$, there exists a vector chart $t = (U, \varphi, E)$ of $A$ at $b_0$ as in 7.3.2, a vector chart $t' = (U, \varphi', L)$ of $M$ at $b_0$, and a module structure of $E$ on $L$ such that, for every $b \in U$, $t'_b$ is a $t_b$-isomorphism of the $A_b$-module $M_b$ onto the $E$-module $L$.

7.3.4. Let $A$ be a Banach algebra over $K$ (for example a field endowed with a finite-dimensional $K$-algebra structure). The trivial fibré $A_B$ is then a fibré in algebras, locally trivial. A locally trivial fibré in $A_B$-modules $M$ is also called a vector fibré over $A$ (with base $B$). The fibres $M_b$ are then topological $A$-modules. An $f$-morphism $u$ of $M$ into another vector fibré over $A$ is said to be $A$-linear if the mappings $u_b$ are $A$-linear for every $b \in B$.

7.4. Sections

7.4.1. Let $M$ be a vector fibré with base $B$. For every open $U$ of $B$, we denote by $\mathscr{S}_M^r(U)$ the set of sections of class $C^r$ of $M$ over $U$, that is to say, morphisms $s$ of class $C^r$ from $U$ into $M$ such that $s(b) \in M_b$ for every $b \in U$. This set is endowed with a module structure over the ring $\mathscr{C}^r(U)$ of morphic functions by the rules:

(1)

7.4.2. Let $M_1, \ldots, M_d$ and $N$ be vector bundles with base $M$ and let $u$ be a multilinear morphism from $M_1 \times_B \ldots \times_B M_d$ into $N$. For $1 \leq j \leq d$, let us give ourselves a section $s_j$ of $M_j$ over an open set $U$ of $B$; one defines a section $u(s_1, \ldots, s_d)$ of $N$ over $U$ by the formula:

The mapping $(s_1, \ldots, s_d) \mapsto u(s_1, \ldots, s_d)$ is $\mathscr{C}^r(U)$-multilinear. It is sometimes denoted $\mathscr{S}(u)$.

7.4.3. Let $f$ be a morphism of a manifold $B'$ into $B$ and let $M$ be a vector bundle with base $B$. For every open set $U$ of $B$ and every $s \in \mathscr{S}_M^r(U)$, the mapping $x \mapsto (x, s(f(x)))$ is a section of class $C^r$ of $f^*M$ over the open set $f^{-1}(U)$, denoted $f^*s$ and called the inverse image of $s$ by $f$. The mapping $s \mapsto f^*s$ from $\mathscr{S}_M^r(U)$ into $\mathscr{S}_{f^*M}(f^{-1}(U))$ is semilinear with respect to the homomorphism $g \mapsto g \circ (f|f^{-1}(U))$ of $\mathscr{C}^r(U)$ into $\mathscr{C}^r(f^{-1}(U))$.

If moreover $N$ is a vector bundle with base $B'$ and $g$ an $f$-comorphism of $M$ into $N$, one sometimes denotes by $\mathscr{S}(g)$ the mapping $s \mapsto g \circ f^*s$ from $\mathscr{S}_M^r(U)$ into $\mathscr{S}_N^r(f^{-1}(U))$.

7.4.4. Let $M$ be a vector bundle with base $B$, of finite rank. A frame of $M$ over an open subset $U$ of $B$ is called a finite sequence $(s_1, \ldots, s_n)$ of sections of $M$ over $U$ such that $(s_1(b), \ldots, s_n(b))$ is a basis of the vector space $M_b$ for every $b \in B$. The sequence $(s_1, \ldots, s_n)$ is then a basis of the $\mathscr{C}^r(U)$-module $\mathscr{S}_M^r(U)$. If $f$ is a morphism of a manifold $B'$ into $B$, the sections $f^*s_j$ form a frame of $f^*M$ over $f^{-1}(U)$.

7.4.5. Let $L$ be a field, endowed with a structure of $K$-algebra of finite dimension and let $(M, B, \pi)$ be a fibration. Suppose that on each fibre $M_b$ a structure of vector space over $L$, of finite dimension, is given. There exists then at most one structure of vector bundle over $L$ with base $B$ on $M$, compatible with the mapping $\pi$, the manifold structure of $M$ and the $L$-vector space structures on the fibres (7.3.4). For such a structure to exist, it is necessary and sufficient that the following condition be satisfied:

(FV) For every $b_0 \in B$, there exists an open neighbourhood $U$ of $b_0$ in $B$ and an isomorphism of manifolds $\varphi$ from $\pi^{-1}(U)$ onto the product $U \times F$ of $U$ by a vector space $F$ over $L$ of finite dimension, such that for every $b \in U$, the bijection $\varphi_b$ of $M_b$ onto $F$ induced by $\varphi$ is an isomorphism of vector spaces over the field $L$.

The triples $(U, \varphi, F)$ satisfying (FV) are then vector bundle charts of the vector bundle $M$.

The condition (FV) is equivalent to: (FV') For every $b_0 \in B$, there exists an integer $n$ and $n$ sections $s_1, \ldots, s_n$ of $M$ over an open neighbourhood $U$ of $b_0$ such that the mapping

7.4.6. Let $M_1, \ldots, M_d$ and $N$ be vector bundles with base $B$, the $M_j$ being of finite rank. Suppose that for every open set $U$ of $B$ a mapping $\varphi_U$ from $\mathscr{S}_{M_1}^r(U) \times \cdots \times \mathscr{S}_{M_d}^r(U)$ into $\mathscr{S}_N^r(U)$ is given, $\mathscr{C}^r(U)$-multilinear, such that for $V \subset U$ one has:

There then exists one and only one multilinear morphism $u$ from $M_1 \times_B \ldots \times_B M_d$ into $N$ such that $\varphi_U(s_1, \ldots, s_d) = u(s_1, \ldots, s_d)$ whatever the sections $s_j$ of $M$ on the open set $U$ of $B$.

7.4.7. Let $f$ be a morphism of a variety $B'$ into $B$ and let $M$ (resp. $M'$) be a vector bundle with base $B$ (resp. $B'$) and of finite rank. Suppose that for every open set $U$ of $B$ a mapping $\varphi_U$ from $\mathscr{S}_M^r(U)$ into $\mathscr{S}_{M'}^{r'}(f^{-1}(U)), \mathscr{C}^r(U)$-semilinear, is given, such that

for every open set $V \subset U$. There then exists one and only one $f$-comorphism $g$ from $M$ into $M'$ such that $\varphi_U(s) = \mathscr{S}(g)(s)$ for every $s \in \mathscr{S}_M^r(U)$.

*7.4.8. Let $\mathscr{F}$ be a sheaf of modules over the sheaf of rings $\mathscr{C}_B^r$. One says that $\mathscr{F}$ is locally free if for every $b \in B$, there exists an open neighbourhood $U$ of $b$ and an integer $n$ such that $\mathscr{F}|U$ is isomorphic (as a sheaf of $\mathscr{C}_U^r$-modules) to the sheaf $(\mathscr{C}_U^r)^n$.

If $M$ is a vector bundle of base $B$ of finite rank, the sheaf $\mathscr{S}_M^r$ is locally free. Conversely, for every locally free sheaf $\mathscr{F}$ on $B$, there exists a vector bundle $M$ and an isomorphism of sheaves from $\mathscr{S}_M^r$ onto $\mathscr{F}$. If $M'$ is another vector bundle of base $B$ of finite rank, the mapping $g \mapsto \mathscr{S}(g)$ is a bijection from the set of $B$-morphisms of $M$ into $M'$ onto the set of morphisms of sheaves of $\mathscr{C}^r$-modules from $\mathscr{S}_M^r$ into $\mathscr{S}_{M'}^{r'}$*

7.5. Vector subbundles, quotient vector bundles, exact sequences

In this number, a vector bundle means a vector bundle of base $B$ and a morphism of vector bundles means an $\mathrm{Id}_B$-morphism.

7.5.1. Let $M$ be a vector bundle. A subset $M'$ of $M$ is called a vector subbundle of $M$ if, for every point $b \in B$, there exists a vector chart $t = (U, \varphi, E)$ of $M$ at $b$ and a closed vector subspace $F$ of $E$ admitting a topological supplement, such that

Under these conditions, there exists on $M'$ one and only one structure of vector bundle for which the canonical injection of $M'$ into $M$ is a morphism. For each $b \in B$, the fibre $M'_b$ of $M'$ is the closed vector subspace $M' \cap M_b$ of $M_b$; $M'$ is a closed submanifold of $M$ and the structure of underlying manifold of the structure of vector bundle of $M'$ coincides with that induced by the manifold structure of $M$.

7.5.2. Let $M'$ be a vector subbundle of $M$. Let $R\{x, y\}$ denote the following relation between points $x, y$ of $M$: « there exists an element $b$ of $B$ such that $x \in M_b, y \in M_b$ and $x - y \in M'_b$ ». Then $R$ is a regular equivalence relation on $M$ (cf. n° 5.9.7). On the set $M/R$, there exists one and only one structure of vector bundle such that the canonical mapping of $M$ onto $M/R$ is a morphism. We denote it by $M/M'$ and call the vector bundle thus defined the quotient of $M$ by $M'$; for each point $b$ of $B$, the bundle $(M/M')_b$ is the quotient topological vector space $M_b/M'_b$ and the manifold structure on $M/M'$ is the quotient of that of $M$.

7.5.3. Retain the hypotheses of 7.5.2. For every point $b_0$ of $B$, one can find an open neighbourhood $U$ of $b_0$, a Banach space $F$, direct sum of two closed subspaces $F'$ and $F''$, and an isomorphism of vector bundles $\iota$ from $F_U$ onto $M|U$ with the following properties: (i) The restriction $\iota'$ of $\iota$ to $U \times F'$ is an isomorphism from $F'_U$ onto $M'|U$. (ii) If $\rho$ is the canonical mapping of $M$ onto $M'' = M/M'$, and if $\iota''$ is the restriction of $\iota$ to $U \times F''$, the map $\rho \circ \iota''$ is an isomorphism from $F''_U$ onto $M''|U$.

7.5.4. If a morphism of vector bundles $g : L \to M$ has its image contained in $M'$, it is a morphism of vector bundles from $L$ into $M'$. Consider now a morphism of vector bundles $h : M \to N$ and suppose that, for every $b$ in $B$, the restriction of $h_b$ to $M'_b$ is zero. If $\rho$ is the canonical morphism from $M$ onto $M/M'$, there exists one and only one morphism $\bar{h}$ from $M/M'$ into $N$ such that $h = \bar{h} \circ \rho$.

7.5.5. Let $P$ and $Q$ be two vector bundles, and let $g$ be a morphism from $P$ into $Q$; for every point $b$ of $B$, denote by $N_b$ and $I_b$ respectively the kernel and the image of the linear mapping $g_b : P_b \to Q_b$. Put $N = \bigcup_{b \in B} N_b$ and $I = \bigcup_{b \in B} I_b$. The morphism $g$ is said to be locally direct if $N$ is a vector subbundle of $P$ and $I$ a vector subbundle of $Q$. The morphism $g$ then defines by passing to the quotient an isomorphism of $P/N$ onto $I$. It is said that $N$ is the kernel of $g$ and it is denoted by $\mathrm{Ker}\,g$. Analogously, the subbundle $I$ is called the image of $g$ and is denoted by $\mathrm{Im}\,g$. If $r \geqslant \infty$, the morphism $g$ is locally direct if and only if $g$ is a subimmersion. If $P$ is of finite rank, the morphism $g$ is locally direct if and only if the vector rank of $g$ is locally constant or equivalently if and only if $g$ is a subimmersion.

7.5.6. Let $M \xrightarrow{f} M' \xrightarrow{g} M''$ be two morphisms of vector bundles. We say that the sequence $(f, g)$ is locally direct exact if the two morphisms $f$ and $g$ are locally direct and if $\operatorname{Im} f = \operatorname{Ker} g$. If $g \circ f = 0$, the set $D$ of points $b \in B$ such that the sequence $M_b \xrightarrow{f_b} M'_b \xrightarrow{g_b} M''_b$ is direct exact (that is to say, such that $\operatorname{Ker} f_b$ and $\operatorname{Im} g_b$ admit topological complements and that $\operatorname{Im} f_b = \operatorname{Ker} g_b$) is open and the sequence $M|D \xrightarrow{f} M'|D \xrightarrow{g} M''|D$ is locally direct exact.

One defines analogously the locally direct exact sequences of arbitrary length. By abuse of language, one sometimes says direct exact sequence instead of locally direct exact sequence.

7.5.7. Let $0 \to M \xrightarrow{f} M' \xrightarrow{g} M'' \to 0$ be a sequence of morphisms of vector bundles. For this sequence to be locally direct exact, it is necessary and sufficient that $f$ be an isomorphism of $M$ onto a vector subbundle $f(M)$ of $M'$ and that $g$ define by passing to the quotient an isomorphism of the quotient vector bundle $M'/f(M)$ onto $M''$.

7.6. Vector functors

In this no. and in the following three no.s 7.7 to 7.9, the letter I denotes a finite set, union of two disjoint subsets $I_+$ and $I_-$. We denote by $\mathscr{V} = (V_i)_{i \in I}$ (and analogously by $\mathscr{V}', \mathscr{V}'', \ldots$) a family of Banach spaces indexed by $I$. We denote by $\operatorname{Hom}(\mathscr{V}, \mathscr{V}')$ the Banach space $\prod_{i \in I_+} \mathscr{L}(V_i; V'_i) \times \prod_{i \in I_-} \mathscr{L}(V'_i; V_i)$ and by $f = (f_i)$ an element of $\operatorname{Hom}(\mathscr{V}, \mathscr{V}')$. We denote by $\operatorname{Id}_{\mathscr{V}}$ the element $(\operatorname{Id}_{V_i})_{i \in I}$ of $\operatorname{Hom}(\mathscr{V}, \mathscr{V})$. For $f \in \operatorname{Hom}(\mathscr{V}, \mathscr{V}')$ and $f' \in \operatorname{Hom}(\mathscr{V}', \mathscr{V}'')$, we denote by $f' \circ f$ the element of $\operatorname{Hom}(\mathscr{V}, \mathscr{V}'')$ whose components are given by: