Closed and finite subspace [duplicate]
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This question already has an answer here:
Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
3 answers
Given $X$ a normed space, $M$ a closed subspace of $X$ and $Z$ a finite dimension subspace of $X.$ Show that $M+Z$ is a closed subspace of $X.$
How can I do this? I thought to construct a sequence convergent in $M+Z$ with limit in $M+Z$ but I don’t know if it is the right way and anyway I don’t know how to construct this sequence...
sequences-and-series functional-analysis banach-spaces normed-spaces closed-graph
edited Dec 6 '18 at 23:56
user376343
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asked Dec 6 '18 at 13:56
Maggie94Maggie94
856
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marked as duplicate by Martin Sleziak, GNUSupporter 8964民主女神 地下教會, Davide Giraudo functional-analysis
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Dec 9 '18 at 14:44
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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$begingroup$
This question already has an answer here:
Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
3 answers
Given $X$ a normed space, $M$ a closed subspace of $X$ and $Z$ a finite dimension subspace of $X.$ Show that $M+Z$ is a closed subspace of $X.$
How can I do this? I thought to construct a sequence convergent in $M+Z$ with limit in $M+Z$ but I don’t know if it is the right way and anyway I don’t know how to construct this sequence...
sequences-and-series functional-analysis banach-spaces normed-spaces closed-graph
edited Dec 6 '18 at 23:56
user376343
3,3582826
asked Dec 6 '18 at 13:56
Maggie94Maggie94
856
$endgroup$
marked as duplicate by Martin Sleziak, GNUSupporter 8964民主女神 地下教會, Davide Giraudo functional-analysis
Users with the functional-analysis badge can single-handedly close functional-analysis questions as duplicates and reopen them as needed.
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Dec 9 '18 at 14:44
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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See also: Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
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– Martin Sleziak
Dec 9 '18 at 13:05
add a comment |
$begingroup$
This question already has an answer here:
Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
3 answers
Given $X$ a normed space, $M$ a closed subspace of $X$ and $Z$ a finite dimension subspace of $X.$ Show that $M+Z$ is a closed subspace of $X.$
How can I do this? I thought to construct a sequence convergent in $M+Z$ with limit in $M+Z$ but I don’t know if it is the right way and anyway I don’t know how to construct this sequence...
sequences-and-series functional-analysis banach-spaces normed-spaces closed-graph
edited Dec 6 '18 at 23:56
user376343
3,3582826
asked Dec 6 '18 at 13:56
Maggie94Maggie94
856
$endgroup$
This question already has an answer here:
Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
3 answers
Given $X$ a normed space, $M$ a closed subspace of $X$ and $Z$ a finite dimension subspace of $X.$ Show that $M+Z$ is a closed subspace of $X.$
How can I do this? I thought to construct a sequence convergent in $M+Z$ with limit in $M+Z$ but I don’t know if it is the right way and anyway I don’t know how to construct this sequence...
This question already has an answer here:
Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
3 answers
sequences-and-series functional-analysis banach-spaces normed-spaces closed-graph
sequences-and-series functional-analysis banach-spaces normed-spaces closed-graph
edited Dec 6 '18 at 23:56
user376343
3,3582826
asked Dec 6 '18 at 13:56
Maggie94Maggie94
856
edited Dec 6 '18 at 23:56
user376343
3,3582826
asked Dec 6 '18 at 13:56
Maggie94Maggie94
856
edited Dec 6 '18 at 23:56
user376343
3,3582826
edited Dec 6 '18 at 23:56
user376343
3,3582826
edited Dec 6 '18 at 23:56
user376343
3,3582826
3,3582826
asked Dec 6 '18 at 13:56
Maggie94Maggie94
856
asked Dec 6 '18 at 13:56
Maggie94Maggie94
856
asked Dec 6 '18 at 13:56
Maggie94Maggie94
856
856
marked as duplicate by Martin Sleziak, GNUSupporter 8964民主女神 地下教會, Davide Giraudo functional-analysis
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Dec 9 '18 at 14:44
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Martin Sleziak, GNUSupporter 8964民主女神 地下教會, Davide Giraudo functional-analysis
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Dec 9 '18 at 14:44
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
See also: Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
$endgroup$
– Martin Sleziak
Dec 9 '18 at 13:05
add a comment |
$begingroup$
See also: Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
$endgroup$
– Martin Sleziak
Dec 9 '18 at 13:05
$begingroup$
See also: Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
$endgroup$
– Martin Sleziak
Dec 9 '18 at 13:05
$begingroup$
See also: Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
$endgroup$
– Martin Sleziak
Dec 9 '18 at 13:05
add a comment |
1 Answer
1
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Note that $X / M$ is also a normed space with the quotient norm, because $M$ was assumed to be closed. Define the projection $pi colon X rightarrow X setminus M$, which is continuous. Now note that $W:= pi(Z)$ is a finite dimensional subspace of $X /M$. Thus $M+ Z = pi^{-1}(W)$ is a closed subspace.
answered Dec 6 '18 at 14:24
p4schp4sch
4,995217
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Wow, easier than I thought. Thank you very much!! Can you also help me with this? math.stackexchange.com/questions/3029768/…
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– Maggie94
Dec 7 '18 at 11:58
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Note that $X / M$ is also a normed space with the quotient norm, because $M$ was assumed to be closed. Define the projection $pi colon X rightarrow X setminus M$, which is continuous. Now note that $W:= pi(Z)$ is a finite dimensional subspace of $X /M$. Thus $M+ Z = pi^{-1}(W)$ is a closed subspace.
answered Dec 6 '18 at 14:24
p4schp4sch
4,995217
$endgroup$
$begingroup$
Wow, easier than I thought. Thank you very much!! Can you also help me with this? math.stackexchange.com/questions/3029768/…
$endgroup$
– Maggie94
Dec 7 '18 at 11:58
add a comment |
$begingroup$
Note that $X / M$ is also a normed space with the quotient norm, because $M$ was assumed to be closed. Define the projection $pi colon X rightarrow X setminus M$, which is continuous. Now note that $W:= pi(Z)$ is a finite dimensional subspace of $X /M$. Thus $M+ Z = pi^{-1}(W)$ is a closed subspace.
answered Dec 6 '18 at 14:24
p4schp4sch
4,995217
$endgroup$
$begingroup$
Wow, easier than I thought. Thank you very much!! Can you also help me with this? math.stackexchange.com/questions/3029768/…
$endgroup$
– Maggie94
Dec 7 '18 at 11:58
add a comment |
$begingroup$
Note that $X / M$ is also a normed space with the quotient norm, because $M$ was assumed to be closed. Define the projection $pi colon X rightarrow X setminus M$, which is continuous. Now note that $W:= pi(Z)$ is a finite dimensional subspace of $X /M$. Thus $M+ Z = pi^{-1}(W)$ is a closed subspace.
answered Dec 6 '18 at 14:24
p4schp4sch
4,995217
$endgroup$
Note that $X / M$ is also a normed space with the quotient norm, because $M$ was assumed to be closed. Define the projection $pi colon X rightarrow X setminus M$, which is continuous. Now note that $W:= pi(Z)$ is a finite dimensional subspace of $X /M$. Thus $M+ Z = pi^{-1}(W)$ is a closed subspace.
answered Dec 6 '18 at 14:24
p4schp4sch
4,995217
answered Dec 6 '18 at 14:24
p4schp4sch
4,995217
answered Dec 6 '18 at 14:24
p4schp4sch
4,995217
answered Dec 6 '18 at 14:24
p4schp4sch
4,995217
4,995217
$begingroup$
Wow, easier than I thought. Thank you very much!! Can you also help me with this? math.stackexchange.com/questions/3029768/…
$endgroup$
– Maggie94
Dec 7 '18 at 11:58
add a comment |
$begingroup$
Wow, easier than I thought. Thank you very much!! Can you also help me with this? math.stackexchange.com/questions/3029768/…
$endgroup$
– Maggie94
Dec 7 '18 at 11:58
$begingroup$
Wow, easier than I thought. Thank you very much!! Can you also help me with this? math.stackexchange.com/questions/3029768/…
$endgroup$
– Maggie94
Dec 7 '18 at 11:58
$begingroup$
Wow, easier than I thought. Thank you very much!! Can you also help me with this? math.stackexchange.com/questions/3029768/…
$endgroup$
– Maggie94
Dec 7 '18 at 11:58
add a comment |
$begingroup$
See also: Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
$endgroup$
– Martin Sleziak
Dec 9 '18 at 13:05