The  canonical  line  bundle on $\mathbb{C}P^{n}$ is  denoted by $\ell_{n}$. It is  well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic  bundles  as two  topological vector  bundles over $\mathbb{C}P^{1}$.

>(A topological question) How can we  compare $(\ell_{n}\otimes \ell_{n})\oplus1 $  with $\ell_{n}\oplus\ell_{n}$. We know  that their  restriction to $\mathbb{C}P^{1}\subset \mathbb{C}P^{n}$ are isomorphic. So what is  the  **difference element**  as  an  element of the relative  K-theory $K(\mathbb{C}P^{n},\mathbb{C}P^{1})$?


>(An  algebraic  geometric  question) Is the  identity $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ valid  as  line bundles in the  context  of  algebraic geometry?