What is the $K$-theory of the category of coherent sheaves on the infinite (countable) dimensional projective space over a field? As far as I know, $K$-theory is oriented; hence this theory should be "simple" (power series in one variable over the $K$-theory of the base field). Yet I am not sure that this answer gives those $K$-groups that I am interested in. So, I would be deeply grateful for an explanation!

What is the $K$-theory of the category of coherent sheaves on the infinite (countable) dimensional projective space over a field? As far as I know, $K$-theory is oriented; hence this theory should be "simple" (power series in one variable over the $K$-theory of the base field). Yet I am not sure that this answer gives those $K$-groups that I am interested in. Also, does the answer depend on the choice of a definition of the infinite dimensional projective space (how many possible definitions are there?)? I would be deeply grateful for an explanation!