Don't really understand this question. So is it a linear map?

Yes. Given (integrable?) functions u, x, and constant a, recall that we can define the functions u + x and au as:

(u + v)(x) = u(x) + v(x)

(au)(x) = a u(x)

Then by properties of integrals, we have:

f(u + v) = \int_0^1 (u(x) + v(x)) dx = \int_0^1 u(x) dx + \int_0^1 v(x)) dx = f(u) + f(v)

f(au) = \int_0^1 a u(x) dx = a \int_0^1 u(x) dx = a f(u)

It is a linear map.