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Consider the sequence space $\ell^p=L^p(\mathbb N)$, where $1\leq p \leq \infty$. Let $\mathbf x_n=(x^n_k)_{k=1}^\infty$ be a sequence in $\ell^p$, such that $\mathbf x_n\overset{w}{\to} \mathbf x$. The question is: what additional condition are required to imply that $\mathbf x_n$ converges to $\mathbf x$ in norm?

The case $p=1$ does not require any additional conditions, as answered here, but for $1<p<\infty$, the sequence $\mathbf e_n=(0,0,\ldots,0,1,0,,0\ldots)$ is $w$-convergent but not convergent in norm.

So, is there a nice sufficient condition for $\mathbf x_n$ to be convergent in norm, expressed in terms of, for example, a bound?

Ma Joad
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  • A very strong condition is that there be an $x$ in $\ell^p$ with $|x^{(n)}|≥ |x_k^{(n)}|$ for all $n,k$, ie an $x$ that dominates the sequence. – s.harp May 29 '20 at 09:29

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The additional condition is $$\tag{*} \lim_{K\to +\infty}\sup_{n\geqslant 1}\sum_{k\geqslant K}\lvert x_{k}^n-x^n\rvert^p=0. $$ Indeed, write $$ \sum_{k\geqslant 1}\left\lvert x_{k}^n-x_n\right\rvert^p=\sum_{k=1}^{K-1}\left\lvert x_{k}^n-x_n\right\rvert^p+\sum_{k\geqslant K}\left\lvert x_{k}^n-x_n\right\rvert^p\leqslant \sum_{k=1}^{K-1}\left\lvert x_{k}^n-x_n\right\rvert^p+\sup_{N\geqslant 1}\sum_{k\geqslant K}\left\lvert x_{k}^N-x^N\right\rvert^p. $$ Using the weak convergence, we derive that for all fixed $K$, $\sum_{k=1}^{K-1}\left\lvert x_{k}^n-x_n\right\rvert^p\to 0$ hence $$ \limsup_{n\to \infty}\lVert x_n-x\rVert_p\leqslant \sup_{N\geqslant 1}\sum_{k\geqslant K}\left\lvert x_{k}^N-x^N\right\rvert^p. $$

Davide Giraudo
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