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I've recently been browsing some Berkeley exams and I'm particularly interested in Problem 19 here.

Let ${f_n}$ be a sequence of continuous real functions defined on $[0,1]$ such that $\int_0^1 (f_n(y))^2 dy \le 5$ for all $n$.

Define $g_n : [0,1] \rightarrow \mathbb{R}$ by $g_n(x) = \int ^1_0 \sqrt{x + y} f_{n} (y)dy$.

  1. Find a constant $K > 0$ such that $|g_n(x)| \le K$ for all $n$.
  2. Prove that a subsequence of the sequence $\{g_n\}$ converges uniformly.

Could you help me solve it?

Frankly speaking, I don't know how to use the condition that $\int_0^1 (f_n(y))^2 dy \le 5$.

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  1. By the Cauchy-Schwarz Inequality we have $$ \left|g_n(x)\right| \leqslant \sqrt{\int_0^1 (x+y) \, dy } \sqrt{\int_0^1 (f_n(y))^2 \, dy } \leqslant \sqrt{\int_0^1 (1+y) \, dy }\, \sqrt{5} = \sqrt{\frac{15}{2}} $$

Alternatively, we can also use the fact that $\left|f_n (x)\right| \leqslant 1 + (f_n(x))^2$ and see that $$\int_0^1 \left|f_n(x)\right| \, dx \leqslant 1 + 5 = 6,$$ so $$\left|g_n(x)\right| \leqslant \sqrt{2} \int_0^1 \left|f_n(y)\right| dy \leqslant 6\sqrt{2}.$$

  1. Since $\sqrt{x+y}$ is a continuous function of $x$ and $y$ on the compact unit square, it is uniformly continuous there. Hence, given any $\varepsilon > 0$, there exists $\delta > 0$ such that $$\left|\sqrt{x_1 + y_1} - \sqrt{x_2 + y_2}\right| < \varepsilon $$ whenever $\left|x_1 - x_2\right| + \left|y_1 - y_2 \right| < \delta$. In particular, $\left|\sqrt{x_1 + y} - \sqrt{x_2 + y}\right| < \varepsilon$ whenever $\left|x_1 - x_2\right| < \delta$, therefore, \begin{eqnarray*} \left|g_{n} (x_1) - g_{n} (x_2) \right|& =& \left|\int_0^1 \left(\sqrt{x_1+y} - \sqrt{x_2 + y} \right) f_n(y) \, dy \right|\\ & \leqslant &\varepsilon \int_0^1 \left|f_n(y)\right| \, dy \\ &\leqslant &5 \varepsilon \end{eqnarray*} whenever $\left|x_1 - x_2\right| < \delta$. Since the same value of $\delta$ works for all values of $n$ simultaneously, the family $\{ g_n \}$ is equicontinuous. Using the uniform bound established above the conclusion follows from the Arzelà-Ascoli's Theorem.
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Looks like a Cauchy-Schwarz problem in $L^2$; have you tried that approach?

For part 2, I would try Arzela-Ascoli, since we have boundedness from part 1.

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