To start with, you need to be clear about what the space C0(0,1) is, and what the norm on it is. I think you probably mean the space that I would call C[0,1], which consists of all continuous functions on the closed interval [0,1]. The norm of a function f in this space is , the maximum value of |f| on the interval.

The operator L takes a function to its integral: L(f) is the function whose value at x is . What you need to do is to see whether is less than some multiple of . So you want to estimate the maximum value of . This must be less then or equal to . But for all s, by the definition of .

Now see if you can take it from there and show that for some constant c.