About Locally Compact Groups and Analysis
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This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed.
Let $G$ be a locally compact group and let $f in C_c(G)$ where $C_c(G)$ is the set of all continuous functions on $G$ with compact supports. Then $forall epsilon > 0$, there is an open neighborhood $U$ of the identity such that whenever $y in xU$, it follows that $|f(x) - f(y)| < epsilon$.
What is the identity? what is a locally compact group? and what they mean for $xU$ ? a very odd problem for an analysis qualifying.
real-analysis measure-theory continuity locally-compact-groups
edited Jan 10 at 17:08
David C. Ullrich
61.7k43995
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,086314
$endgroup$
add a comment |
$begingroup$
This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed.
Let $G$ be a locally compact group and let $f in C_c(G)$ where $C_c(G)$ is the set of all continuous functions on $G$ with compact supports. Then $forall epsilon > 0$, there is an open neighborhood $U$ of the identity such that whenever $y in xU$, it follows that $|f(x) - f(y)| < epsilon$.
What is the identity? what is a locally compact group? and what they mean for $xU$ ? a very odd problem for an analysis qualifying.
real-analysis measure-theory continuity locally-compact-groups
edited Jan 10 at 17:08
David C. Ullrich
61.7k43995
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,086314
$endgroup$
1
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
add a comment |
$begingroup$
This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed.
Let $G$ be a locally compact group and let $f in C_c(G)$ where $C_c(G)$ is the set of all continuous functions on $G$ with compact supports. Then $forall epsilon > 0$, there is an open neighborhood $U$ of the identity such that whenever $y in xU$, it follows that $|f(x) - f(y)| < epsilon$.
What is the identity? what is a locally compact group? and what they mean for $xU$ ? a very odd problem for an analysis qualifying.
real-analysis measure-theory continuity locally-compact-groups
edited Jan 10 at 17:08
David C. Ullrich
61.7k43995
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,086314
$endgroup$
This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed.
Let $G$ be a locally compact group and let $f in C_c(G)$ where $C_c(G)$ is the set of all continuous functions on $G$ with compact supports. Then $forall epsilon > 0$, there is an open neighborhood $U$ of the identity such that whenever $y in xU$, it follows that $|f(x) - f(y)| < epsilon$.
What is the identity? what is a locally compact group? and what they mean for $xU$ ? a very odd problem for an analysis qualifying.
real-analysis measure-theory continuity locally-compact-groups
real-analysis measure-theory continuity locally-compact-groups
edited Jan 10 at 17:08
David C. Ullrich
61.7k43995
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,086314
edited Jan 10 at 17:08
David C. Ullrich
61.7k43995
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,086314
edited Jan 10 at 17:08
David C. Ullrich
61.7k43995
edited Jan 10 at 17:08
David C. Ullrich
61.7k43995
edited Jan 10 at 17:08
David C. Ullrich
61.7k43995
61.7k43995
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,086314
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,086314
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,086314
1,086314
1
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
add a comment |
1
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
1
1
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
add a comment |
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$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09