# if a subset in $C([0,1])$ is bounded and equicontinuous

Let $$F= \{ f \in C([0,1]) : \int_0^1 |f(x)| dx \leq 1 \}$$ then does $$F$$ bounded and equicontuous?

I am not sure how can I relate the function values with the integral of the function.

• Are you asking if the set $F$ is bounded with respect to the supremum norm? Jan 22, 2021 at 16:12
• @CameronWilliams, yes, under the sup norm. Jan 22, 2021 at 16:13
• You know that $$f_n(x) = x^n$$ is not equicontinuous. How would you modify this? Jan 22, 2021 at 16:14
• Can you construct a sequence of positive functions that have integral $1$ but have norms increasing to infinity? Hint: consider a triangle that has height $2n$ and width $1/n$. Jan 22, 2021 at 16:14
• @CameronWilliams. I remember these functions, I constructed them their areas are 1, and they are not bounded, but are they not equicontinuous, correct? Jan 22, 2021 at 17:50

Consider $$f_n(x)=(n+1)x^n$$
Notice that for $$x \neq 1$$, $$|f_n(1) - f_n(x)| = (n+1)(1 - x_0^n)$$ and as $$x^n \to 0$$ for $$x \neq 1$$, we see that the difference $$|f_n(1) - f_n(x)|$$ will not be bounded as $$n$$ came large, therefore this family is not equicontinuous.