Why the lower limit of the integral is 2 and upper one is zero? Anyway:

if <f,f> = 0 then both the integral and ||f(1)|| = 0 . Now, you didn't say but if we're working with continuous functions on [0,2] then a non-negative function's integral on [0,2] equals zero iff the function is zero, because otherwise, by continuity, there'd be an inteval (a,b) where f is not zero ==> IN{0,2}(||f(x)||^2)dx >= INT{a,b}(||f(x)||^2 dx > 0, by properties of Riemann's Integral.

If you're not working with continuous functions then the above indeed isn't an inner product

Tonio