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Compact set on $L^p(0,1)$ space
The following is the question in my textbook:
Let $0 < p \leq \infty$ and let $X$ be the space of $L^p$ functions $f$ that satisfy the integral equation:
$$
f(x) = \left( \int_0^1f(x-y)g(y)dy \right)g(x)
$$
for some fixed $g \in L^q$ with $1/p + 1/q = 1$. Suppose $g$ does not decay too fast at zero.
Show that the closed unit ball in $X$ is compact.
My attempt:
By Banach-Alaoglu, we can find a subsequence $f_{n_k}$ of $f_n$ converging weakly to some $f$. If the limit is the zero function, the problem is solved. Otherwise, the net $g_{n_k}(x) = \left( \int_0^1f_{n_k}(x-y)g(y)dy \right)g_{n_k}(x)$ converges pointwise to some function, say $h$. To show that $h = g$, I used the fact that $h(x) = h(1)g(x)$ and the triangle inequality to conclude that $h(x) = g(x)$ for a.e. $x \in [0,1]$. But from this I can only conclude that $f_{n_k}$ converges weakly to a function $f$ such that $f(x) = g(x)$. I am also struggling to show that $f$ is continuous. Any
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// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
package org.chromium.chrome.browser.download;
import org.chromium.chrome.browser.download.AutoDeleteDownloadManager;
/**
* Browser Download Manger used by DownloadJob.
*/
public interface DownloadManager {
/**
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* When auto-delete is started, any new downloads start downloading to a temporary location and then
* deleted when they finish. In most cases, this is the same location the
* download was created in.
*/
void startAutoDelete();
/**
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*/
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boolean isAutoDeleteStarted();
/**
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*/
AutoDeleteDownloadManager getAutoDeleteManager();
/**
* Gets the DownloadTask for the specified download.
*/
DownloadTask getDownloadTask(DownloadInfo downloadInfo);
/**
* Cancels the specified DownloadTask.
*/
void cancelDownloadTask(DownloadTask task);
/**
* Starts the specified DownloadTask.
*
* @param downloadInfo the download info for the download task to start.
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