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?
Why is $L^3$ weaker than $L^2$?
2
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metric-spaces
convergence
normed-spaces
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1That would depend on what $A_n,B_n$ would be. – 2012-10-29
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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
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0Presumably, you are getting at this: http://math.stackexchange.com/questions/21460/how-to-show-that-lp-spaces-are-nested – 2012-10-29