# Show that sequence is uniformly bounded and equicontinuous.

Let $$K: [0,1] \times [0,1] \to \mathbb{R}$$ be continuous. Let $$f_n : [0,1] \to \mathbb{R}^n$$ be continuous and uniformly bounded. Define

$$g_n(x) = \int_{0}^{1} K(x,y) f_n(y) \ dy, \ \ x \in [0,1], n \geq 1.$$ Show that the sequence $$\{g_n(x) \}$$ is uniformly bounded and equicontinuous.

Attempt:

To show that the sequence is uniformly bounded, we must show there exists an $$M$$ such that $$|g_n(x)| \leq M,\ \forall n \in \mathbb{N}$$ and for all x.

Since $$f_n(x)$$ are uniformly bounded, we have that $$|f_n(x)| \leq M'$$. Since $$K(x,y)$$ is continuous on a compact interval, we have that $$K(x,y)$$ reaches a maximum := $$N$$. Thus, $$|g_n(x)| = \left|\int_{0}^{1} K(x,y) f_n(y) \ dy\right| \leq \left|M' \int_{0}^{1} K(x,y) \ dy\right| \leq |M'N(1-0)| := M.$$ Thus, the sequence $$\{g_n(x)\}$$ is uniformly bounded.

To show that the sequence is equicontinuous, we must show that for every $$\epsilon > 0$$ there exists a $$\delta$$ (which can depend on $$x_0$$) such that $$|g_n(x) - g_n(x_0)| < \epsilon$$ when $$|x-x_0| < \delta.$$ Well, $$|g_n(x) - g_n(x_0)| = \left|\int_{0}^{1} K(x,y) f_n(y) \ dy - \int_{0}^{1} K(x_0,y) f_n(y) \ dy\right| \leq M' \left|\int_{0}^{1} K(x,y) - K(x_0, y) \ dy \right|.$$

Since $$K$$ is uniformly continuous (because $$K$$ is continuous on a compact set), we have that $$|K(x,y) - K(x_0, y)| < \epsilon/M'$$ whenever $$|x-x_0| < \delta$$.

So, we get $$|g_n(x) - g_n(x_0)| \leq M' \left|\int_{0}^{1} K(x,y) - K(x_0, y) \ dy \right| \leq M' \int_{0}^{1} |K(x,y) - K(x_0, y)| \ dy \leq \ M'\epsilon/M' = \epsilon$$ whenever $$|x-x_0| < \delta.$$ Hence, the sequence is equicontinuous.

• You dropped absolute values in the first part. And you have a mysterious $y_0$ in the second part.
– zhw.
Apr 17, 2017 at 16:19

Hint: Use the uniform continuity of $K$ on $[0,1]\times [0,1].$
• That looks correct. You still have some housekeeping to do, like introduce $\epsilon$ earlier, tend to a few more absolute value signs. And is $[0,1]\times [0,1]$ an interval?