2
$\begingroup$

Someone told me today that if I can show $\Vert A_n-B_n\Vert_3\to 0$ as $n\to \infty$, then claiming $A=B$ as $n\to \infty$ (where $A$ and $B$ are the respective limits of $A_n$ and $B_n$) is a weaker claim than if I were to show that $\Vert A_n-B_n\Vert_2\to 0$ (which in turn is weaker than if I were to show $\Vert A_n-B_n\Vert_1\to 0$). Why is this so?

  • 1
    That would depend on what $A_n,B_n$ would be.2012-10-29
  • 0
    @tomasz I don't follow... if need be, ignore $A_n$ and $B_n$ and just consider $\Vert A-B\Vert_3$ (and similarly, 2 and 1)2012-10-29
  • 0
    Presumably, you are getting at this: http://math.stackexchange.com/questions/21460/how-to-show-that-lp-spaces-are-nested2012-10-29

1 Answers 1