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Results tagged with fa.functional-analysis
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user 75500
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
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votes
1
answer
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Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?
For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p_\omega$-norm of the sequence $v$ by
$$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right …
2
votes
Accepted
Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?
The answer to both questions is no.
Taking $q=1$, $p=2$, $v_k = k^{-(1+\varepsilon)}$ and $\omega_k = k^{(1+3/2\varepsilon)/2}$ for some $\varepsilon>0$ provides a counter-example to the first assert …