Norm of linear continuous functions on a Banach space
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I have this lemma, is it correct like this or i must say $$sup_{langle g,yrangle=0, |y|=1}|langle f,yrangle|=min_{lambdainmathbb{R}}|f-lambda g|$$
functional-analysis analysis banach-spaces
edited Dec 5 '18 at 7:42
p4sch
4,885217
asked Apr 15 '18 at 13:30
VrouvrouVrouvrou
1,9191722
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add a comment |
$begingroup$
I have this lemma, is it correct like this or i must say $$sup_{langle g,yrangle=0, |y|=1}|langle f,yrangle|=min_{lambdainmathbb{R}}|f-lambda g|$$
functional-analysis analysis banach-spaces
edited Dec 5 '18 at 7:42
p4sch
4,885217
asked Apr 15 '18 at 13:30
VrouvrouVrouvrou
1,9191722
$endgroup$
add a comment |
$begingroup$
I have this lemma, is it correct like this or i must say $$sup_{langle g,yrangle=0, |y|=1}|langle f,yrangle|=min_{lambdainmathbb{R}}|f-lambda g|$$
functional-analysis analysis banach-spaces
edited Dec 5 '18 at 7:42
p4sch
4,885217
asked Apr 15 '18 at 13:30
VrouvrouVrouvrou
1,9191722
$endgroup$
I have this lemma, is it correct like this or i must say $$sup_{langle g,yrangle=0, |y|=1}|langle f,yrangle|=min_{lambdainmathbb{R}}|f-lambda g|$$
functional-analysis analysis banach-spaces
functional-analysis analysis banach-spaces
edited Dec 5 '18 at 7:42
p4sch
4,885217
asked Apr 15 '18 at 13:30
VrouvrouVrouvrou
1,9191722
edited Dec 5 '18 at 7:42
p4sch
4,885217
asked Apr 15 '18 at 13:30
VrouvrouVrouvrou
1,9191722
edited Dec 5 '18 at 7:42
p4sch
4,885217
edited Dec 5 '18 at 7:42
p4sch
4,885217
edited Dec 5 '18 at 7:42
p4sch
4,885217
4,885217
asked Apr 15 '18 at 13:30
VrouvrouVrouvrou
1,9191722
asked Apr 15 '18 at 13:30
VrouvrouVrouvrou
1,9191722
asked Apr 15 '18 at 13:30
VrouvrouVrouvrou
1,9191722
1,9191722
add a comment |
add a comment |
1 Answer
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If $X$ is a complex banach space, you will need to take the absolute value, since $langle f,y rangle$ is maybe a complex number. On the other, if $X$ is real, everything is okay! (One obvious reason is that we can replace $y$ by $-y$ if $langle f,y rangle$ is negative.)
answered Apr 15 '18 at 13:45
p4schp4sch
4,885217
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add a comment |
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1 Answer
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1 Answer
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active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
If $X$ is a complex banach space, you will need to take the absolute value, since $langle f,y rangle$ is maybe a complex number. On the other, if $X$ is real, everything is okay! (One obvious reason is that we can replace $y$ by $-y$ if $langle f,y rangle$ is negative.)
answered Apr 15 '18 at 13:45
p4schp4sch
4,885217
$endgroup$
add a comment |
$begingroup$
If $X$ is a complex banach space, you will need to take the absolute value, since $langle f,y rangle$ is maybe a complex number. On the other, if $X$ is real, everything is okay! (One obvious reason is that we can replace $y$ by $-y$ if $langle f,y rangle$ is negative.)
answered Apr 15 '18 at 13:45
p4schp4sch
4,885217
$endgroup$
add a comment |
$begingroup$
If $X$ is a complex banach space, you will need to take the absolute value, since $langle f,y rangle$ is maybe a complex number. On the other, if $X$ is real, everything is okay! (One obvious reason is that we can replace $y$ by $-y$ if $langle f,y rangle$ is negative.)
answered Apr 15 '18 at 13:45
p4schp4sch
4,885217
$endgroup$
If $X$ is a complex banach space, you will need to take the absolute value, since $langle f,y rangle$ is maybe a complex number. On the other, if $X$ is real, everything is okay! (One obvious reason is that we can replace $y$ by $-y$ if $langle f,y rangle$ is negative.)
answered Apr 15 '18 at 13:45
p4schp4sch
4,885217
answered Apr 15 '18 at 13:45
p4schp4sch
4,885217
answered Apr 15 '18 at 13:45
p4schp4sch
4,885217
answered Apr 15 '18 at 13:45
p4schp4sch
4,885217
4,885217
add a comment |
add a comment |
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