(see also Chern-Weil theory, parameterized homotopy theory)
For $E_1, E_2 \to X$ two vector bundles, their direct sum over $X$, also called their Whitney sum, is the vector bundle $E_1 \oplus E_2 \to X$ whose fiber over any $x \in X$ is the direct sum of vector spaces of the fibers of $E_1$ and $E_2$.
(direct sum of topological vector bundles via total spaces)
Let
$X$ be a topological space,
$E_1 \overset{\pi_1}{\to} X$ and $E_2 \overset{\pi_2}{\to} X$ two topological vector bundles over $X$.
Then the direct sum of vector bundles $E_1 \oplus_X E_2 \to E$ is the topological vector bundle whose total space is the topological subspace
of the product topological space of the two total spaces, and whose projection map is
For $x \in X$ the vector space structure on the fibers
is the one on the direct sum of vector spaces.
(direct sum of topological vector bundles via transition functions)
Let $X$ be a topological space, and let $E_1 \to X$ and $E_2 \to X$ be two topological vector bundles over $X$.
Let $\{U_i \subset X\}_{i \in I}$ be an open cover with respect to which both vector bundles locally trivialize (this always exists: pick a local trivialization of either bundle and form the joint refinement of the respective open covers by intersection of their patches). Let
be the transition functions of these two bundles with respect to this cover.
For $i, j \in I$ write
be the pointwise direct sum of these transition functions
Then the direct sum bundle $E_1 \oplus E_2$ is the one glued from this direct sum of the transition functions (by this construction):
Let $X$ and $Y$ be topological spaces, and write $X \sqcup Y$ for their disjoint union space.
Then every topological vector bundle on $X \sqcup Y$ is the direct sum of a vector bundle that has rank zero on $Y$ and one that has rank zero on $X$.
More explicitiy: let
and
be the operations of extending a vector bundle on the other connected component by a rank-0 vector bundle, then
is an isomorphism of isomorphism classes of vector bundles (and an equivalence of categories of categories of vector bundles before passing to isomorphism classes).
(sub-bundles over paracompact spaces are direct summands)
Let
$X$ be a paracompact Hausdorff space,
$E \to X$ a topological vector bundle.
Then every vector subbundle $E_1 \hookrightarrow E$ is a direct vector bundle summand, in that there exists another vector subbundle $E_2 \hookrightarrow E$ such that their direct sum of vector bundles (def. 1) is $E$
Since $X$ is assumed to be paracompact Hausdorff, there exists a inner product on vector bundles
(by this prop.). This defines at each $x \in X$ the orthogonal complement $(E'_x)^\perp \subset E_x$ of $E'_x \hookrightarrow E$. The subspace of these orthogonal complements is readily checked to be a topological vector bundle $(E')^\perp \to X$. Hence by construction we have
(over compact Hausdorff spaces every vector bundle is direct summand of a trivial bundle)
Let
$X$ be a compact Hausdorff space;
$E \to X$ a topological vector bundle.
Then there exists another topological vector bundle $\tilde E \to X$ such that the direct sum of vector bundles (def. 1) of the two is a trivial vector $X \times \mathbb{R}^n$:
(e.g. Hatcher, prop. 1.4, Friedlander, ptop. 3.1)
Let $\{U_i \subset X\}_{i \in I}$ be an open cover of $X$ over which $E \to X$ has a local trivialization
By compactness of $X$, there is a finite sub-cover, hence a finite set $J \subset I$ such tat
is still an open cover over which $E$ trivializes.
Since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity there exists a partition of unity
with support $supp(f_i) \subset U_i$. Hence the functions
extend by 0 to vector bundle homomorphism of the form
The finite pointwise direct sum of these yields a vector bundle homomorphism of the form
Observe that, as opposed to the single $f_i \cdot \phi^{-1}_i$, this is a fiber-wise injective, because at each point at least one of the $f_i$ is non-vanishing. Hence this is an injection of $E$ into a trivial vector bundle.
With this the statement follows by prop. 1.
Prop. 2 is key for the construction of topological K-theory groups on compact Hausdorff spaces.
Discussion with an eye towards topological K-theory is in
Max Karoubi, K-theory. An introduction, Grundlehren der Mathematischen Wissenschaften 226, Springer 1978. xviii+308 pp.
Allen Hatcher, section 1.1 of Vector bundles and K-Theory, (partly finished book) web
and with an eye towards algebraic K-theory in