I was reading Michael Atiyah's lectures on K-theory and I came across this paragraph which I couldn't understand.
So we have a compact Hausdorff space $X$ and a vector bundle $\pi : E \rightarrow X $ . Let $P(E)$ denote the projectivization of the vector bundle with projection $p : P(E) \rightarrow X$ . Then Atiyah says "any $a\in P(E)_x=P(E_x)$ represents a one dimensional subspace $H^*_x \subset E_x $. The union of all these define a subspace $H^* \subset p^*E$ which turns out to be a sub-bundle."
As far as I understand $p^*E$ is a vector bundle over $P(E)$. So I guess $H^*$ should be defined as for $a \in P(E) \\ $ , $H^*_a = \{ v \in p^*E_a = E_{p(a)} \&\ v \in a \} $. Am I correct ? I am new to K theory and would like to get more insights in such type of constructions.
Yeah, you are correct. Given a bundle $\pi:E \to B$ you can form the projectivization $q:\mathbb P(E) \to B$, and pullback along $q$ to get a bundle over $\mathbb P(E)$. In particular this bundle is given by
$$q^*(E)=\{(e,\ell) \in E \times \mathbb P(E) \mid q(\ell)=p(e) \}.$$
note that this condition is satisfied by $e \in \ell$, and hence we have the "tautological sub-bundle"
$$\{(e, \ell) \in E \times \mathbb P(E) \mid e \in \ell \} \to\mathbb P(E)$$
which happens to be one dimensional over $\mathbb P(E)$ since it consists of all points in $E$ contained in some line $\ell$.