8
votes
3answers
670 views

Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$

Hi. Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I mean by this ...
10
votes
1answer
629 views

Non-probabilistic proof of the Johnson–Lindenstrauss lemma

The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...