I'm reading Bott and Tu's Differential Forms in Algebraic Topology and the discussion they give for constructing direct sums, tensor products, and duals of vector bundles is too quick for me to get a good understanding. If anyone can check if my understanding below is correct it would be appreciated.
From what I can gleam, if $M$ is a smooth manifold with vector bundles $E$ and $E'$, then $$E \oplus E' = \bigsqcup_{x \in M}E_{x} \oplus E'_{x} \qquad E \otimes E' = \bigsqcup_{x \in M}E_{x} \otimes E'_{x} \qquad E^{*} = \bigsqcup_{x \in M}E^{*}_{x}$$ as sets with the standard topology induced by the inclusion and then disjoint union where $E_{x}$ and $E'_{x}$ are the fibers over $x$ for $E$ and $E'$ respectively. The associated continuous surjective map $\pi$ into $M$ is given by mapping each element to its index in the disjoint union.
My main gap in understanding how the trivializations for these new spaces are continuous and smooth. Let me give an example with the direct sum. If $\{\phi_{\alpha}\}$ and $\{\phi'_{\alpha}\}$ are trivializations for $E$ and $E'$, then the glued trivialization map should be $$\phi_{\alpha} \oplus \phi'_{\alpha}: E \oplus E'|_{U_{\alpha}} \to U_{\alpha} \times (\mathbb{R}^{n} \oplus \mathbb{R}^{m}) \qquad (e_{x},e'_{x}) \mapsto (x,\phi_{\alpha,2}(e_{x}),\phi'_{\alpha,2}(e'_{x}))$$ where the subscripts $2$ indicate the second component of the map. Obviously this map is bijective. To see it is a diffeomorphism notice that $$E \oplus E' \cong \bigsqcup_{x \in M} \{x\} \oplus E_{x} \oplus E'_{x}.$$ Then we just need to show the function $(x,e_{x},e'_{x}) \mapsto (x,\phi_{\alpha,2}(e_{x}),\phi'_{\alpha,2}(e'_{x}))$ is a diffeomorphism. But it is in each component after restricting to the fiber of $x$ (we can do this because the domain is the disjoint union over all fibers) so we are done.
Is this correct (for the direct sum case) or am I missing anything? This would also generalize to the other two operations mentioned, right?
This is not correct. Your descriptions of the maps are correct, but your description of the manifold structure on $E\oplus E'$ is totally wrong, and as a result so is your explanation for why the maps are diffeomorphisms (and in any case your explanation for that part would not be right even if you had the correct manifold structure). The formula $$E \oplus E' = \bigsqcup_{x \in M}E_{x} \oplus E'_{x}$$ describes $E\oplus E'$ as a set, but not as a topological space or as a smooth manifold: the fibers $E_x\oplus E'_x$ are not just disconnected pieces put together with the disjoint union topology, but are joined together similar to how the fibers $E_x$ are joined together to form $E$ (which is a $(\dim M+ \dim E_x)$-dimensional manifold, not just a disjoint union of $\dim E_x$-dimensional vector spaces). In particular, you cannot test smoothness of a map on $E\oplus E'$ by just looking at each individual fiber, any more than you can test smoothness of a map on $\mathbb{R}^2$ by testing that it is smooth on each vertical line.
So, how do you do this correctly? What you do instead is you define the smooth structure on $E\oplus E'$ to be such that your maps $\phi_\alpha\oplus \phi_\alpha'$ are diffeomorphisms. In other words, you take those maps to be charts on $E\oplus E'$ which you use to define its manifold structure. To be sure that this is valid, you need to test that when two different charts overlap, they are compatible. The transition function between two such charts will turn out to be just built out of the transition functions of the vector bundles $E$ and $E'$, and so will be smooth since those transition functions are smooth. (Indeed, if you think of vector bundles as being given by a system of transition functions, there is a much simpler description of the construction: just take the vector bundle whose transition functions are the direct sum of transition functions for $E$ and transition functions for $E'$.)
The same ideas apply to the other constructions: you define the smooth structure on $E\otimes E'$ and $E^*$ such that the maps you want to be trivializations are diffeomorphisms. (Or alternatively, you can again construct transition functions for the vector bundle as tensor products/duals of the original transition functions.)