Mixing
Proof on a generalization of Hensel's lifting lemma
$begingroup$
I am reading a proof on the generalization of Hensel's lifting lemma (over p-adic integers), particularly on 1-lipschitz functions. I came across a part with notations which I've never encountered before, hence I got confused.
Here are the important given for that part:
$bullet M_k subset {0,1,2,...,p^k-1} , k geq1\
bullet h_i in M_i \
bullet mu text{ is the set of all possible sequences } h_1, h_2,h_3,...,h_k text{ where } h_kin M_k $
And then the confusing part goes like this:
Since $$lim_gets M_k = mu$$
and since $f$ is a continuous function,
then, $$f(a)= 0, forall a in mu$$
I don't get the symbol with the limit with the left arrow, and how that implies to the conclusion shown.
I would appreciate if someone could make this clearer to me. Thank you!
p-adic-number-theory
edited Jan 5 at 10:53
Bernard
119k740113
asked Jan 5 at 9:41
Jonelle YuJonelle Yu
1776
$endgroup$
add a comment |
$begingroup$
I am reading a proof on the generalization of Hensel's lifting lemma (over p-adic integers), particularly on 1-lipschitz functions. I came across a part with notations which I've never encountered before, hence I got confused.
Here are the important given for that part:
$bullet M_k subset {0,1,2,...,p^k-1} , k geq1\
bullet h_i in M_i \
bullet mu text{ is the set of all possible sequences } h_1, h_2,h_3,...,h_k text{ where } h_kin M_k $
And then the confusing part goes like this:
Since $$lim_gets M_k = mu$$
and since $f$ is a continuous function,
then, $$f(a)= 0, forall a in mu$$
I don't get the symbol with the limit with the left arrow, and how that implies to the conclusion shown.
I would appreciate if someone could make this clearer to me. Thank you!
p-adic-number-theory
edited Jan 5 at 10:53
Bernard
119k740113
asked Jan 5 at 9:41
Jonelle YuJonelle Yu
1776
$endgroup$
1
$begingroup$
It is the standard notation for the inverse (or projective) limit.
$endgroup$
– Bernard
Jan 5 at 10:57
$begingroup$
"$f$ is a continuous function" from which to which topological space?
$endgroup$
– Torsten Schoeneberg
Jan 5 at 17:37
$begingroup$
Sorry, from the set of p adic integers to itself
$endgroup$
– Jonelle Yu
Jan 8 at 3:38
add a comment |
$begingroup$
I am reading a proof on the generalization of Hensel's lifting lemma (over p-adic integers), particularly on 1-lipschitz functions. I came across a part with notations which I've never encountered before, hence I got confused.
Here are the important given for that part:
$bullet M_k subset {0,1,2,...,p^k-1} , k geq1\
bullet h_i in M_i \
bullet mu text{ is the set of all possible sequences } h_1, h_2,h_3,...,h_k text{ where } h_kin M_k $
And then the confusing part goes like this:
Since $$lim_gets M_k = mu$$
and since $f$ is a continuous function,
then, $$f(a)= 0, forall a in mu$$
I don't get the symbol with the limit with the left arrow, and how that implies to the conclusion shown.
I would appreciate if someone could make this clearer to me. Thank you!
p-adic-number-theory
edited Jan 5 at 10:53
Bernard
119k740113
asked Jan 5 at 9:41
Jonelle YuJonelle Yu
1776
$endgroup$
I am reading a proof on the generalization of Hensel's lifting lemma (over p-adic integers), particularly on 1-lipschitz functions. I came across a part with notations which I've never encountered before, hence I got confused.
Here are the important given for that part:
$bullet M_k subset {0,1,2,...,p^k-1} , k geq1\
bullet h_i in M_i \
bullet mu text{ is the set of all possible sequences } h_1, h_2,h_3,...,h_k text{ where } h_kin M_k $
And then the confusing part goes like this:
Since $$lim_gets M_k = mu$$
and since $f$ is a continuous function,
then, $$f(a)= 0, forall a in mu$$
I don't get the symbol with the limit with the left arrow, and how that implies to the conclusion shown.
I would appreciate if someone could make this clearer to me. Thank you!
p-adic-number-theory
p-adic-number-theory
edited Jan 5 at 10:53
Bernard
119k740113
asked Jan 5 at 9:41
Jonelle YuJonelle Yu
1776
edited Jan 5 at 10:53
Bernard
119k740113
asked Jan 5 at 9:41
Jonelle YuJonelle Yu
1776
edited Jan 5 at 10:53
Bernard
119k740113
edited Jan 5 at 10:53
Bernard
119k740113
edited Jan 5 at 10:53
Bernard
119k740113
119k740113
asked Jan 5 at 9:41
Jonelle YuJonelle Yu
1776
asked Jan 5 at 9:41
Jonelle YuJonelle Yu
1776
asked Jan 5 at 9:41
Jonelle YuJonelle Yu
1776
1776
1
$begingroup$
It is the standard notation for the inverse (or projective) limit.
$endgroup$
– Bernard
Jan 5 at 10:57
$begingroup$
"$f$ is a continuous function" from which to which topological space?
$endgroup$
– Torsten Schoeneberg
Jan 5 at 17:37
$begingroup$
Sorry, from the set of p adic integers to itself
$endgroup$
– Jonelle Yu
Jan 8 at 3:38
add a comment |
1
$begingroup$
It is the standard notation for the inverse (or projective) limit.
$endgroup$
– Bernard
Jan 5 at 10:57
$begingroup$
"$f$ is a continuous function" from which to which topological space?
$endgroup$
– Torsten Schoeneberg
Jan 5 at 17:37
$begingroup$
Sorry, from the set of p adic integers to itself
$endgroup$
– Jonelle Yu
Jan 8 at 3:38
1
1
$begingroup$
It is the standard notation for the inverse (or projective) limit.
$endgroup$
– Bernard
Jan 5 at 10:57
$begingroup$
It is the standard notation for the inverse (or projective) limit.
$endgroup$
– Bernard
Jan 5 at 10:57
$begingroup$
"$f$ is a continuous function" from which to which topological space?
$endgroup$
– Torsten Schoeneberg
Jan 5 at 17:37
$begingroup$
"$f$ is a continuous function" from which to which topological space?
$endgroup$
– Torsten Schoeneberg
Jan 5 at 17:37
$begingroup$
Sorry, from the set of p adic integers to itself
$endgroup$
– Jonelle Yu
Jan 8 at 3:38
$begingroup$
Sorry, from the set of p adic integers to itself
$endgroup$
– Jonelle Yu
Jan 8 at 3:38
add a comment |
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1
$begingroup$
It is the standard notation for the inverse (or projective) limit.
$endgroup$
– Bernard
Jan 5 at 10:57
$begingroup$
"$f$ is a continuous function" from which to which topological space?
$endgroup$
– Torsten Schoeneberg
Jan 5 at 17:37
$begingroup$
Sorry, from the set of p adic integers to itself
$endgroup$
– Jonelle Yu
Jan 8 at 3:38