Integer times vector bundle notation in Hatcher

Question

In Hatcher's vector bundles and K-theory, p45, there is a proposition written "If $q$ is a polynomial clutching function of degree at most $n$, then $[E,q] \oplus [nE,\mathbb{1}] \approx [(n+1)E,L^nq]$ for a linear clutching function $L^nq$.

Here $E$ is a vector bundle and $[E,f]$ denotes the vector bundle constructed by $E$ and the clutching function $f$. However I have no clue what $nE$ or $(n+1)E$ mean, and naturally no definition or explanation of the notation is given. The proof of the proposition mentions that $(n+1)E$ has direct summands, so I thought it might be the direct sum (interpreted appropriately for vector bundles) of $E$ $n+1$ times, but I am not sure so I was hoping someone who understands this notation could confirm this.

Answer 1

It's only semi-true at best that "naturally no definition or explanation of the notation is given," see the first paragraph under "Ring Structure" on page 40:

We can simplify notation by writing for the element $\epsilon^n \in K(X)$ just as $n$. This is consistent with familiar arithmetic rules. For example, the product $n E$ is the sum of $n$ copies of $E$.

In other words, it is the tensor product $\epsilon^n \otimes E \cong \bigoplus_{i = 1}^n E$.

I say semi-true because this definition is technically given on the level of classes, not on the level of bundles; but really this shouldn't pose a problem, and going back and forth tacitly between these two things is common in the literature. Also, do keep in mind that this is not a finished, published book, so don't be too harsh on it :)

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If you're just here for the punchline: Yes, your interpretation of the notation is correct.