Riesz lemma for $L^p$ space
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I need a proof for special case of Riesz lemma (when $varepsilon$ is 0):
If Y is a closed proper subspace of $L^p(mu)$ for some $1<p<infty$, then there exist $fin L^p(mu)$ such that $||f||=1$ and $||f-g||geq 1$ for every $gin Y$.
I know that Clarkson's inequality (uniform convexity) can be used.
functional-analysis banach-spaces lp-spaces
asked Jan 12 at 10:19
HanaHana
161
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add a comment |
$begingroup$
I need a proof for special case of Riesz lemma (when $varepsilon$ is 0):
If Y is a closed proper subspace of $L^p(mu)$ for some $1<p<infty$, then there exist $fin L^p(mu)$ such that $||f||=1$ and $||f-g||geq 1$ for every $gin Y$.
I know that Clarkson's inequality (uniform convexity) can be used.
functional-analysis banach-spaces lp-spaces
asked Jan 12 at 10:19
HanaHana
161
$endgroup$
$begingroup$
This is an easy application of Hahn-Banach (hint: Every element of the dual of $L^p$ for $1<p<infty$ attains its norm...)
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– David C. Ullrich
Jan 12 at 13:06
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Can you give me further explanation how to apply that?
$endgroup$
– Hana
Jan 16 at 11:08
add a comment |
$begingroup$
I need a proof for special case of Riesz lemma (when $varepsilon$ is 0):
If Y is a closed proper subspace of $L^p(mu)$ for some $1<p<infty$, then there exist $fin L^p(mu)$ such that $||f||=1$ and $||f-g||geq 1$ for every $gin Y$.
I know that Clarkson's inequality (uniform convexity) can be used.
functional-analysis banach-spaces lp-spaces
asked Jan 12 at 10:19
HanaHana
161
$endgroup$
I need a proof for special case of Riesz lemma (when $varepsilon$ is 0):
If Y is a closed proper subspace of $L^p(mu)$ for some $1<p<infty$, then there exist $fin L^p(mu)$ such that $||f||=1$ and $||f-g||geq 1$ for every $gin Y$.
I know that Clarkson's inequality (uniform convexity) can be used.
functional-analysis banach-spaces lp-spaces
functional-analysis banach-spaces lp-spaces
asked Jan 12 at 10:19
HanaHana
161
asked Jan 12 at 10:19
HanaHana
161
asked Jan 12 at 10:19
HanaHana
161
asked Jan 12 at 10:19
HanaHana
161
asked Jan 12 at 10:19
HanaHana
161
161
$begingroup$
This is an easy application of Hahn-Banach (hint: Every element of the dual of $L^p$ for $1<p<infty$ attains its norm...)
$endgroup$
– David C. Ullrich
Jan 12 at 13:06
$begingroup$
Can you give me further explanation how to apply that?
$endgroup$
– Hana
Jan 16 at 11:08
add a comment |
$begingroup$
This is an easy application of Hahn-Banach (hint: Every element of the dual of $L^p$ for $1<p<infty$ attains its norm...)
$endgroup$
– David C. Ullrich
Jan 12 at 13:06
$begingroup$
Can you give me further explanation how to apply that?
$endgroup$
– Hana
Jan 16 at 11:08
$begingroup$
This is an easy application of Hahn-Banach (hint: Every element of the dual of $L^p$ for $1<p<infty$ attains its norm...)
$endgroup$
– David C. Ullrich
Jan 12 at 13:06
$begingroup$
This is an easy application of Hahn-Banach (hint: Every element of the dual of $L^p$ for $1<p<infty$ attains its norm...)
$endgroup$
– David C. Ullrich
Jan 12 at 13:06
$begingroup$
Can you give me further explanation how to apply that?
$endgroup$
– Hana
Jan 16 at 11:08
$begingroup$
Can you give me further explanation how to apply that?
$endgroup$
– Hana
Jan 16 at 11:08
add a comment |
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$begingroup$
This is an easy application of Hahn-Banach (hint: Every element of the dual of $L^p$ for $1<p<infty$ attains its norm...)
$endgroup$
– David C. Ullrich
Jan 12 at 13:06
$begingroup$
Can you give me further explanation how to apply that?
$endgroup$
– Hana
Jan 16 at 11:08