Banach space convergence [closed]
Having $X$ , a Banach space. Show that ${x_n}$ converging to $x$ implies that for all functions $f$ contained in $X^ast$ (dual), $f(x_n)$ converges to $f(x)$.
banach-spaces
closed as off-topic by Kavi Rama Murthy, Shailesh, Christopher, Martin Sleziak, Gibbs Nov 23 at 17:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Kavi Rama Murthy, Shailesh, Christopher, Martin Sleziak, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
Having $X$ , a Banach space. Show that ${x_n}$ converging to $x$ implies that for all functions $f$ contained in $X^ast$ (dual), $f(x_n)$ converges to $f(x)$.
banach-spaces
closed as off-topic by Kavi Rama Murthy, Shailesh, Christopher, Martin Sleziak, Gibbs Nov 23 at 17:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Kavi Rama Murthy, Shailesh, Christopher, Martin Sleziak, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
1
Sequential definition of continuity makes this entirely trivial. Note that even linearity of $f$ need not be explicitly used.
– Kavi Rama Murthy
Nov 23 at 7:24
See Wikipedia entry en.wikipedia.org/wiki/Banach_space
– Kavi Rama Murthy
Nov 23 at 7:35
add a comment |
Having $X$ , a Banach space. Show that ${x_n}$ converging to $x$ implies that for all functions $f$ contained in $X^ast$ (dual), $f(x_n)$ converges to $f(x)$.
banach-spaces
Having $X$ , a Banach space. Show that ${x_n}$ converging to $x$ implies that for all functions $f$ contained in $X^ast$ (dual), $f(x_n)$ converges to $f(x)$.
banach-spaces
banach-spaces
edited Nov 23 at 7:08
Robert Lewis
43.4k22863
43.4k22863
asked Nov 23 at 6:59
mimi
175
175
closed as off-topic by Kavi Rama Murthy, Shailesh, Christopher, Martin Sleziak, Gibbs Nov 23 at 17:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Kavi Rama Murthy, Shailesh, Christopher, Martin Sleziak, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Kavi Rama Murthy, Shailesh, Christopher, Martin Sleziak, Gibbs Nov 23 at 17:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Kavi Rama Murthy, Shailesh, Christopher, Martin Sleziak, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
1
Sequential definition of continuity makes this entirely trivial. Note that even linearity of $f$ need not be explicitly used.
– Kavi Rama Murthy
Nov 23 at 7:24
See Wikipedia entry en.wikipedia.org/wiki/Banach_space
– Kavi Rama Murthy
Nov 23 at 7:35
add a comment |
1
Sequential definition of continuity makes this entirely trivial. Note that even linearity of $f$ need not be explicitly used.
– Kavi Rama Murthy
Nov 23 at 7:24
See Wikipedia entry en.wikipedia.org/wiki/Banach_space
– Kavi Rama Murthy
Nov 23 at 7:35
1
1
Sequential definition of continuity makes this entirely trivial. Note that even linearity of $f$ need not be explicitly used.
– Kavi Rama Murthy
Nov 23 at 7:24
Sequential definition of continuity makes this entirely trivial. Note that even linearity of $f$ need not be explicitly used.
– Kavi Rama Murthy
Nov 23 at 7:24
See Wikipedia entry en.wikipedia.org/wiki/Banach_space
– Kavi Rama Murthy
Nov 23 at 7:35
See Wikipedia entry en.wikipedia.org/wiki/Banach_space
– Kavi Rama Murthy
Nov 23 at 7:35
add a comment |
2 Answers
2
active
oldest
votes
Assuming $X^ast$ denotes the vector space of continuous linear functionals
$f:X to Bbb F, tag 1$
where $Bbb F = Bbb R$ or $Bbb F = Bbb C$ is the base field, then every $f in X^ast$ is bounded, viz,
$exists 0 < C_f in Bbb R, ; vert f(x) - f(y) vert le C_f vert x - y vert, ; forall x, y in X; tag 2$
then if
$x_n to x ; text{as} ; n to infty, tag 3$
given $0 < epsilon in Bbb R$ we have, for $n$ sufficiently large,
$vert x_n - x vert < epsilon; tag 4$
thus, for such $n$,
$vert f(x_n) - f(x) vert le C_f vert x_n - x vert < C_f epsilon; tag 5$
taking $epsilon$ small enough ensures $C_f epsilon$ is itself arbitrarily small, whence we see that (5), by definition, implies that
$f(x_n) to f(x) ; text{as} ; n to infty. tag 6$
For a finite dimension of X , the reciprocal would be true right? Because of isomorphism
– mimi
Nov 27 at 23:22
@mimi: in finite dimensions, yes.
– Robert Lewis
Nov 27 at 23:25
add a comment |
Look how you define the norm of $f$ for $f in V^*$: $$|f|_* = sup_{z neq 0} frac{lvert f(z) rvert}{|z|}.$$ In particular, for any fixed $x in X$, we see $$|f|_* ge frac{lvert f(x) rvert}{|x|} ,,,, implies ,,,,, lvert f(x) rvert le |f|_* |x|.$$ Now suppose that $x_n to x$ in $X$; that is, suppose that $|x_n - x| to 0$. What can you say about $lvert f(x_n) - f(x)rvert?$
I don't understand why the proof has to be so complicated. Isn't this simply continuity of $f$? By definition every element of the dual space is a continuous function.
– Kavi Rama Murthy
Nov 23 at 7:17
1
@KaviRamaMurthy: what is it you find so complicated about this proof? It seems the author has covered the essentials in a failry minimalist manner. Cheers!
– Robert Lewis
Nov 23 at 7:22
I suppose it's fine to just assert that $f$ is continuous. I'm just giving a quantitative measure of continuity here. I don't see that this is particularly complicated; the whole argument concludes in roughly 6 lines.
– User8128
Nov 23 at 7:22
1
@RobertLewis If $f$ is a continuous function on a topological space and $x_n to x$ then $f(x_n) to f(x)$. I don't understand what proof you are asking for. Am i really missing something here?
– Kavi Rama Murthy
Nov 23 at 7:30
1
@RobertLewis The usual definition of the dual of a normed linear space is it is the set of all continuous linear functionals. I havent see any book where the definition of the dual space is given in terms of the operator norm.
– Kavi Rama Murthy
Nov 23 at 7:32
|
show 6 more comments
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Assuming $X^ast$ denotes the vector space of continuous linear functionals
$f:X to Bbb F, tag 1$
where $Bbb F = Bbb R$ or $Bbb F = Bbb C$ is the base field, then every $f in X^ast$ is bounded, viz,
$exists 0 < C_f in Bbb R, ; vert f(x) - f(y) vert le C_f vert x - y vert, ; forall x, y in X; tag 2$
then if
$x_n to x ; text{as} ; n to infty, tag 3$
given $0 < epsilon in Bbb R$ we have, for $n$ sufficiently large,
$vert x_n - x vert < epsilon; tag 4$
thus, for such $n$,
$vert f(x_n) - f(x) vert le C_f vert x_n - x vert < C_f epsilon; tag 5$
taking $epsilon$ small enough ensures $C_f epsilon$ is itself arbitrarily small, whence we see that (5), by definition, implies that
$f(x_n) to f(x) ; text{as} ; n to infty. tag 6$
For a finite dimension of X , the reciprocal would be true right? Because of isomorphism
– mimi
Nov 27 at 23:22
@mimi: in finite dimensions, yes.
– Robert Lewis
Nov 27 at 23:25
add a comment |
Assuming $X^ast$ denotes the vector space of continuous linear functionals
$f:X to Bbb F, tag 1$
where $Bbb F = Bbb R$ or $Bbb F = Bbb C$ is the base field, then every $f in X^ast$ is bounded, viz,
$exists 0 < C_f in Bbb R, ; vert f(x) - f(y) vert le C_f vert x - y vert, ; forall x, y in X; tag 2$
then if
$x_n to x ; text{as} ; n to infty, tag 3$
given $0 < epsilon in Bbb R$ we have, for $n$ sufficiently large,
$vert x_n - x vert < epsilon; tag 4$
thus, for such $n$,
$vert f(x_n) - f(x) vert le C_f vert x_n - x vert < C_f epsilon; tag 5$
taking $epsilon$ small enough ensures $C_f epsilon$ is itself arbitrarily small, whence we see that (5), by definition, implies that
$f(x_n) to f(x) ; text{as} ; n to infty. tag 6$
For a finite dimension of X , the reciprocal would be true right? Because of isomorphism
– mimi
Nov 27 at 23:22
@mimi: in finite dimensions, yes.
– Robert Lewis
Nov 27 at 23:25
add a comment |
Assuming $X^ast$ denotes the vector space of continuous linear functionals
$f:X to Bbb F, tag 1$
where $Bbb F = Bbb R$ or $Bbb F = Bbb C$ is the base field, then every $f in X^ast$ is bounded, viz,
$exists 0 < C_f in Bbb R, ; vert f(x) - f(y) vert le C_f vert x - y vert, ; forall x, y in X; tag 2$
then if
$x_n to x ; text{as} ; n to infty, tag 3$
given $0 < epsilon in Bbb R$ we have, for $n$ sufficiently large,
$vert x_n - x vert < epsilon; tag 4$
thus, for such $n$,
$vert f(x_n) - f(x) vert le C_f vert x_n - x vert < C_f epsilon; tag 5$
taking $epsilon$ small enough ensures $C_f epsilon$ is itself arbitrarily small, whence we see that (5), by definition, implies that
$f(x_n) to f(x) ; text{as} ; n to infty. tag 6$
Assuming $X^ast$ denotes the vector space of continuous linear functionals
$f:X to Bbb F, tag 1$
where $Bbb F = Bbb R$ or $Bbb F = Bbb C$ is the base field, then every $f in X^ast$ is bounded, viz,
$exists 0 < C_f in Bbb R, ; vert f(x) - f(y) vert le C_f vert x - y vert, ; forall x, y in X; tag 2$
then if
$x_n to x ; text{as} ; n to infty, tag 3$
given $0 < epsilon in Bbb R$ we have, for $n$ sufficiently large,
$vert x_n - x vert < epsilon; tag 4$
thus, for such $n$,
$vert f(x_n) - f(x) vert le C_f vert x_n - x vert < C_f epsilon; tag 5$
taking $epsilon$ small enough ensures $C_f epsilon$ is itself arbitrarily small, whence we see that (5), by definition, implies that
$f(x_n) to f(x) ; text{as} ; n to infty. tag 6$
answered Nov 23 at 7:19
Robert Lewis
43.4k22863
43.4k22863
For a finite dimension of X , the reciprocal would be true right? Because of isomorphism
– mimi
Nov 27 at 23:22
@mimi: in finite dimensions, yes.
– Robert Lewis
Nov 27 at 23:25
add a comment |
For a finite dimension of X , the reciprocal would be true right? Because of isomorphism
– mimi
Nov 27 at 23:22
@mimi: in finite dimensions, yes.
– Robert Lewis
Nov 27 at 23:25
For a finite dimension of X , the reciprocal would be true right? Because of isomorphism
– mimi
Nov 27 at 23:22
For a finite dimension of X , the reciprocal would be true right? Because of isomorphism
– mimi
Nov 27 at 23:22
@mimi: in finite dimensions, yes.
– Robert Lewis
Nov 27 at 23:25
@mimi: in finite dimensions, yes.
– Robert Lewis
Nov 27 at 23:25
add a comment |
Look how you define the norm of $f$ for $f in V^*$: $$|f|_* = sup_{z neq 0} frac{lvert f(z) rvert}{|z|}.$$ In particular, for any fixed $x in X$, we see $$|f|_* ge frac{lvert f(x) rvert}{|x|} ,,,, implies ,,,,, lvert f(x) rvert le |f|_* |x|.$$ Now suppose that $x_n to x$ in $X$; that is, suppose that $|x_n - x| to 0$. What can you say about $lvert f(x_n) - f(x)rvert?$
I don't understand why the proof has to be so complicated. Isn't this simply continuity of $f$? By definition every element of the dual space is a continuous function.
– Kavi Rama Murthy
Nov 23 at 7:17
1
@KaviRamaMurthy: what is it you find so complicated about this proof? It seems the author has covered the essentials in a failry minimalist manner. Cheers!
– Robert Lewis
Nov 23 at 7:22
I suppose it's fine to just assert that $f$ is continuous. I'm just giving a quantitative measure of continuity here. I don't see that this is particularly complicated; the whole argument concludes in roughly 6 lines.
– User8128
Nov 23 at 7:22
1
@RobertLewis If $f$ is a continuous function on a topological space and $x_n to x$ then $f(x_n) to f(x)$. I don't understand what proof you are asking for. Am i really missing something here?
– Kavi Rama Murthy
Nov 23 at 7:30
1
@RobertLewis The usual definition of the dual of a normed linear space is it is the set of all continuous linear functionals. I havent see any book where the definition of the dual space is given in terms of the operator norm.
– Kavi Rama Murthy
Nov 23 at 7:32
|
show 6 more comments
Look how you define the norm of $f$ for $f in V^*$: $$|f|_* = sup_{z neq 0} frac{lvert f(z) rvert}{|z|}.$$ In particular, for any fixed $x in X$, we see $$|f|_* ge frac{lvert f(x) rvert}{|x|} ,,,, implies ,,,,, lvert f(x) rvert le |f|_* |x|.$$ Now suppose that $x_n to x$ in $X$; that is, suppose that $|x_n - x| to 0$. What can you say about $lvert f(x_n) - f(x)rvert?$
I don't understand why the proof has to be so complicated. Isn't this simply continuity of $f$? By definition every element of the dual space is a continuous function.
– Kavi Rama Murthy
Nov 23 at 7:17
1
@KaviRamaMurthy: what is it you find so complicated about this proof? It seems the author has covered the essentials in a failry minimalist manner. Cheers!
– Robert Lewis
Nov 23 at 7:22
I suppose it's fine to just assert that $f$ is continuous. I'm just giving a quantitative measure of continuity here. I don't see that this is particularly complicated; the whole argument concludes in roughly 6 lines.
– User8128
Nov 23 at 7:22
1
@RobertLewis If $f$ is a continuous function on a topological space and $x_n to x$ then $f(x_n) to f(x)$. I don't understand what proof you are asking for. Am i really missing something here?
– Kavi Rama Murthy
Nov 23 at 7:30
1
@RobertLewis The usual definition of the dual of a normed linear space is it is the set of all continuous linear functionals. I havent see any book where the definition of the dual space is given in terms of the operator norm.
– Kavi Rama Murthy
Nov 23 at 7:32
|
show 6 more comments
Look how you define the norm of $f$ for $f in V^*$: $$|f|_* = sup_{z neq 0} frac{lvert f(z) rvert}{|z|}.$$ In particular, for any fixed $x in X$, we see $$|f|_* ge frac{lvert f(x) rvert}{|x|} ,,,, implies ,,,,, lvert f(x) rvert le |f|_* |x|.$$ Now suppose that $x_n to x$ in $X$; that is, suppose that $|x_n - x| to 0$. What can you say about $lvert f(x_n) - f(x)rvert?$
Look how you define the norm of $f$ for $f in V^*$: $$|f|_* = sup_{z neq 0} frac{lvert f(z) rvert}{|z|}.$$ In particular, for any fixed $x in X$, we see $$|f|_* ge frac{lvert f(x) rvert}{|x|} ,,,, implies ,,,,, lvert f(x) rvert le |f|_* |x|.$$ Now suppose that $x_n to x$ in $X$; that is, suppose that $|x_n - x| to 0$. What can you say about $lvert f(x_n) - f(x)rvert?$
answered Nov 23 at 7:15
User8128
10.7k1522
10.7k1522
I don't understand why the proof has to be so complicated. Isn't this simply continuity of $f$? By definition every element of the dual space is a continuous function.
– Kavi Rama Murthy
Nov 23 at 7:17
1
@KaviRamaMurthy: what is it you find so complicated about this proof? It seems the author has covered the essentials in a failry minimalist manner. Cheers!
– Robert Lewis
Nov 23 at 7:22
I suppose it's fine to just assert that $f$ is continuous. I'm just giving a quantitative measure of continuity here. I don't see that this is particularly complicated; the whole argument concludes in roughly 6 lines.
– User8128
Nov 23 at 7:22
1
@RobertLewis If $f$ is a continuous function on a topological space and $x_n to x$ then $f(x_n) to f(x)$. I don't understand what proof you are asking for. Am i really missing something here?
– Kavi Rama Murthy
Nov 23 at 7:30
1
@RobertLewis The usual definition of the dual of a normed linear space is it is the set of all continuous linear functionals. I havent see any book where the definition of the dual space is given in terms of the operator norm.
– Kavi Rama Murthy
Nov 23 at 7:32
|
show 6 more comments
I don't understand why the proof has to be so complicated. Isn't this simply continuity of $f$? By definition every element of the dual space is a continuous function.
– Kavi Rama Murthy
Nov 23 at 7:17
1
@KaviRamaMurthy: what is it you find so complicated about this proof? It seems the author has covered the essentials in a failry minimalist manner. Cheers!
– Robert Lewis
Nov 23 at 7:22
I suppose it's fine to just assert that $f$ is continuous. I'm just giving a quantitative measure of continuity here. I don't see that this is particularly complicated; the whole argument concludes in roughly 6 lines.
– User8128
Nov 23 at 7:22
1
@RobertLewis If $f$ is a continuous function on a topological space and $x_n to x$ then $f(x_n) to f(x)$. I don't understand what proof you are asking for. Am i really missing something here?
– Kavi Rama Murthy
Nov 23 at 7:30
1
@RobertLewis The usual definition of the dual of a normed linear space is it is the set of all continuous linear functionals. I havent see any book where the definition of the dual space is given in terms of the operator norm.
– Kavi Rama Murthy
Nov 23 at 7:32
I don't understand why the proof has to be so complicated. Isn't this simply continuity of $f$? By definition every element of the dual space is a continuous function.
– Kavi Rama Murthy
Nov 23 at 7:17
I don't understand why the proof has to be so complicated. Isn't this simply continuity of $f$? By definition every element of the dual space is a continuous function.
– Kavi Rama Murthy
Nov 23 at 7:17
1
1
@KaviRamaMurthy: what is it you find so complicated about this proof? It seems the author has covered the essentials in a failry minimalist manner. Cheers!
– Robert Lewis
Nov 23 at 7:22
@KaviRamaMurthy: what is it you find so complicated about this proof? It seems the author has covered the essentials in a failry minimalist manner. Cheers!
– Robert Lewis
Nov 23 at 7:22
I suppose it's fine to just assert that $f$ is continuous. I'm just giving a quantitative measure of continuity here. I don't see that this is particularly complicated; the whole argument concludes in roughly 6 lines.
– User8128
Nov 23 at 7:22
I suppose it's fine to just assert that $f$ is continuous. I'm just giving a quantitative measure of continuity here. I don't see that this is particularly complicated; the whole argument concludes in roughly 6 lines.
– User8128
Nov 23 at 7:22
1
1
@RobertLewis If $f$ is a continuous function on a topological space and $x_n to x$ then $f(x_n) to f(x)$. I don't understand what proof you are asking for. Am i really missing something here?
– Kavi Rama Murthy
Nov 23 at 7:30
@RobertLewis If $f$ is a continuous function on a topological space and $x_n to x$ then $f(x_n) to f(x)$. I don't understand what proof you are asking for. Am i really missing something here?
– Kavi Rama Murthy
Nov 23 at 7:30
1
1
@RobertLewis The usual definition of the dual of a normed linear space is it is the set of all continuous linear functionals. I havent see any book where the definition of the dual space is given in terms of the operator norm.
– Kavi Rama Murthy
Nov 23 at 7:32
@RobertLewis The usual definition of the dual of a normed linear space is it is the set of all continuous linear functionals. I havent see any book where the definition of the dual space is given in terms of the operator norm.
– Kavi Rama Murthy
Nov 23 at 7:32
|
show 6 more comments
1
Sequential definition of continuity makes this entirely trivial. Note that even linearity of $f$ need not be explicitly used.
– Kavi Rama Murthy
Nov 23 at 7:24
See Wikipedia entry en.wikipedia.org/wiki/Banach_space
– Kavi Rama Murthy
Nov 23 at 7:35