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.
2026-02-23 21:14:33.1771881273
if a subset in $C([0,1])$ is bounded and equicontinuous
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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.