Mixing
Fields closed under countable additions?
Clash Royale CLAN TAG#URR8PPP
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So I was learning about fields today and I came across the property that fields are closed under addition. That is, if $v_2,v_1 in $ F ,then:
$ v_1 + v_2 in$ F
Similarly, I we can say that:
$ v_1 + v_2 +v_3in$ F
My question is can we extend this by saying:
$ sum_n=1^infty v_n in$ F ?
Where $v_n in $ F
I am pretty sure it won't hold for uncountably infinite additions, but will this be true for countably infinite addition as shown?
field-theory
edited Jul 27 at 14:37
Asaf Karagila
291k31402732
asked Jul 27 at 14:27
Alpha Romeo
265313
add a comment |Â
up vote
0
down vote
favorite
So I was learning about fields today and I came across the property that fields are closed under addition. That is, if $v_2,v_1 in $ F ,then:
$ v_1 + v_2 in$ F
Similarly, I we can say that:
$ v_1 + v_2 +v_3in$ F
My question is can we extend this by saying:
$ sum_n=1^infty v_n in$ F ?
Where $v_n in $ F
I am pretty sure it won't hold for uncountably infinite additions, but will this be true for countably infinite addition as shown?
field-theory
edited Jul 27 at 14:37
Asaf Karagila
291k31402732
asked Jul 27 at 14:27
Alpha Romeo
265313
1
How would you define the sum?
– Tobias Kildetoft
Jul 27 at 14:29
4
What does $sum_n=1^infty v_n$ even mean?
– José Carlos Santos
Jul 27 at 14:29
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So I was learning about fields today and I came across the property that fields are closed under addition. That is, if $v_2,v_1 in $ F ,then:
$ v_1 + v_2 in$ F
Similarly, I we can say that:
$ v_1 + v_2 +v_3in$ F
My question is can we extend this by saying:
$ sum_n=1^infty v_n in$ F ?
Where $v_n in $ F
I am pretty sure it won't hold for uncountably infinite additions, but will this be true for countably infinite addition as shown?
field-theory
edited Jul 27 at 14:37
Asaf Karagila
291k31402732
asked Jul 27 at 14:27
Alpha Romeo
265313
So I was learning about fields today and I came across the property that fields are closed under addition. That is, if $v_2,v_1 in $ F ,then:
$ v_1 + v_2 in$ F
Similarly, I we can say that:
$ v_1 + v_2 +v_3in$ F
My question is can we extend this by saying:
$ sum_n=1^infty v_n in$ F ?
Where $v_n in $ F
I am pretty sure it won't hold for uncountably infinite additions, but will this be true for countably infinite addition as shown?
field-theory
edited Jul 27 at 14:37
Asaf Karagila
291k31402732
asked Jul 27 at 14:27
Alpha Romeo
265313
edited Jul 27 at 14:37
Asaf Karagila
291k31402732
edited Jul 27 at 14:37
Asaf Karagila
291k31402732
edited Jul 27 at 14:37
Asaf Karagila
291k31402732
291k31402732
asked Jul 27 at 14:27
Alpha Romeo
265313
asked Jul 27 at 14:27
Alpha Romeo
265313
asked Jul 27 at 14:27
Alpha Romeo
265313
265313
1
How would you define the sum?
– Tobias Kildetoft
Jul 27 at 14:29
4
What does $sum_n=1^infty v_n$ even mean?
– José Carlos Santos
Jul 27 at 14:29
add a comment |Â
1
How would you define the sum?
– Tobias Kildetoft
Jul 27 at 14:29
4
What does $sum_n=1^infty v_n$ even mean?
– José Carlos Santos
Jul 27 at 14:29
1
1
How would you define the sum?
– Tobias Kildetoft
Jul 27 at 14:29
How would you define the sum?
– Tobias Kildetoft
Jul 27 at 14:29
4
4
What does $sum_n=1^infty v_n$ even mean?
– José Carlos Santos
Jul 27 at 14:29
What does $sum_n=1^infty v_n$ even mean?
– José Carlos Santos
Jul 27 at 14:29
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
No. Consider the field $mathbb R$, and the countable sum $sumlimits_n=1^infty1$.
Additional comments:
The problem is that an infinite "sum" is not a sum at all. It is definitely NOT an infinite number of repeated additions. It is the limit of the sequence of finite sums (each of which IS an honest-to-goodness sum, and is an element of the field).
For your sum to exist in an arbitrary field, you would need to be able to define such a limit as a member of that field.
In the counterexample I gave, the partial sums are just $S_k = overbrace1+1+cdots + 1^textrm$k$ terms = k$, but $limlimits_ktoinftyS_k$ does not exist as an element of $mathbb R$.
answered Jul 27 at 14:30
MPW
28.4k11853
That was silly so me, great explanation.
– Alpha Romeo
Jul 27 at 14:35
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
No. Consider the field $mathbb R$, and the countable sum $sumlimits_n=1^infty1$.
Additional comments:
The problem is that an infinite "sum" is not a sum at all. It is definitely NOT an infinite number of repeated additions. It is the limit of the sequence of finite sums (each of which IS an honest-to-goodness sum, and is an element of the field).
For your sum to exist in an arbitrary field, you would need to be able to define such a limit as a member of that field.
In the counterexample I gave, the partial sums are just $S_k = overbrace1+1+cdots + 1^textrm$k$ terms = k$, but $limlimits_ktoinftyS_k$ does not exist as an element of $mathbb R$.
answered Jul 27 at 14:30
MPW
28.4k11853
That was silly so me, great explanation.
– Alpha Romeo
Jul 27 at 14:35
add a comment |Â
up vote
1
down vote
accepted
No. Consider the field $mathbb R$, and the countable sum $sumlimits_n=1^infty1$.
Additional comments:
The problem is that an infinite "sum" is not a sum at all. It is definitely NOT an infinite number of repeated additions. It is the limit of the sequence of finite sums (each of which IS an honest-to-goodness sum, and is an element of the field).
For your sum to exist in an arbitrary field, you would need to be able to define such a limit as a member of that field.
In the counterexample I gave, the partial sums are just $S_k = overbrace1+1+cdots + 1^textrm$k$ terms = k$, but $limlimits_ktoinftyS_k$ does not exist as an element of $mathbb R$.
answered Jul 27 at 14:30
MPW
28.4k11853
That was silly so me, great explanation.
– Alpha Romeo
Jul 27 at 14:35
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
No. Consider the field $mathbb R$, and the countable sum $sumlimits_n=1^infty1$.
Additional comments:
The problem is that an infinite "sum" is not a sum at all. It is definitely NOT an infinite number of repeated additions. It is the limit of the sequence of finite sums (each of which IS an honest-to-goodness sum, and is an element of the field).
For your sum to exist in an arbitrary field, you would need to be able to define such a limit as a member of that field.
In the counterexample I gave, the partial sums are just $S_k = overbrace1+1+cdots + 1^textrm$k$ terms = k$, but $limlimits_ktoinftyS_k$ does not exist as an element of $mathbb R$.
answered Jul 27 at 14:30
MPW
28.4k11853
No. Consider the field $mathbb R$, and the countable sum $sumlimits_n=1^infty1$.
Additional comments:
The problem is that an infinite "sum" is not a sum at all. It is definitely NOT an infinite number of repeated additions. It is the limit of the sequence of finite sums (each of which IS an honest-to-goodness sum, and is an element of the field).
For your sum to exist in an arbitrary field, you would need to be able to define such a limit as a member of that field.
In the counterexample I gave, the partial sums are just $S_k = overbrace1+1+cdots + 1^textrm$k$ terms = k$, but $limlimits_ktoinftyS_k$ does not exist as an element of $mathbb R$.
answered Jul 27 at 14:30
MPW
28.4k11853
edited Jul 27 at 14:37
answered Jul 27 at 14:30
MPW
28.4k11853
answered Jul 27 at 14:30
MPW
28.4k11853
answered Jul 27 at 14:30
MPW
28.4k11853
28.4k11853
That was silly so me, great explanation.
– Alpha Romeo
Jul 27 at 14:35
add a comment |Â
That was silly so me, great explanation.
– Alpha Romeo
Jul 27 at 14:35
That was silly so me, great explanation.
– Alpha Romeo
Jul 27 at 14:35
That was silly so me, great explanation.
– Alpha Romeo
Jul 27 at 14:35
add a comment |Â
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1
How would you define the sum?
– Tobias Kildetoft
Jul 27 at 14:29
4
What does $sum_n=1^infty v_n$ even mean?
– José Carlos Santos
Jul 27 at 14:29