Is it the case that $|f|_{L^p (mathbb{R})} < infty$ iff $|f|_{L^p (mathbb{R})}^p < infty$ for all $1...
Is it the case that, for all $1 leq p < infty$, $$|f|_{L^p (mathbb{R})} < infty quad text{if and only if} quad |f|_{L^p (mathbb{R})}^p < infty?$$
When computing $L^p$ norms, I frequently take the norm to the $p$-th power to make the computation/notation simpler. Is this typically acceptable?
lp-spaces
edited 2 days ago
SvanN
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kkckkc
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Is it the case that, for all $1 leq p < infty$, $$|f|_{L^p (mathbb{R})} < infty quad text{if and only if} quad |f|_{L^p (mathbb{R})}^p < infty?$$
When computing $L^p$ norms, I frequently take the norm to the $p$-th power to make the computation/notation simpler. Is this typically acceptable?
lp-spaces
edited 2 days ago
SvanN
1,8921422
asked 2 days ago
kkckkc
1308
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kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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1
It indeed holds. I would add $1 leq p < infty$ to be more rigorous if you want a statement.
– Rebellos
2 days ago
add a comment |
Is it the case that, for all $1 leq p < infty$, $$|f|_{L^p (mathbb{R})} < infty quad text{if and only if} quad |f|_{L^p (mathbb{R})}^p < infty?$$
When computing $L^p$ norms, I frequently take the norm to the $p$-th power to make the computation/notation simpler. Is this typically acceptable?
lp-spaces
edited 2 days ago
SvanN
1,8921422
asked 2 days ago
kkckkc
1308
New contributor
kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Is it the case that, for all $1 leq p < infty$, $$|f|_{L^p (mathbb{R})} < infty quad text{if and only if} quad |f|_{L^p (mathbb{R})}^p < infty?$$
When computing $L^p$ norms, I frequently take the norm to the $p$-th power to make the computation/notation simpler. Is this typically acceptable?
lp-spaces
lp-spaces
edited 2 days ago
SvanN
1,8921422
asked 2 days ago
kkckkc
1308
New contributor
kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 days ago
SvanN
1,8921422
asked 2 days ago
kkckkc
1308
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kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 2 days ago
SvanN
1,8921422
edited 2 days ago
SvanN
1,8921422
edited 2 days ago
SvanN
1,8921422
1,8921422
asked 2 days ago
kkckkc
1308
New contributor
kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
kkckkc
1308
asked 2 days ago
kkckkc
1308
1308
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kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
It indeed holds. I would add $1 leq p < infty$ to be more rigorous if you want a statement.
– Rebellos
2 days ago
add a comment |
1
It indeed holds. I would add $1 leq p < infty$ to be more rigorous if you want a statement.
– Rebellos
2 days ago
1
1
It indeed holds. I would add $1 leq p < infty$ to be more rigorous if you want a statement.
– Rebellos
2 days ago
It indeed holds. I would add $1 leq p < infty$ to be more rigorous if you want a statement.
– Rebellos
2 days ago
add a comment |
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It indeed holds. I would add $1 leq p < infty$ to be more rigorous if you want a statement.
– Rebellos
2 days ago