A counterintuitive bound on the frobenius norm
In the process of investigating the 
approximation error, I came across an interesting question. From numerical examples, it seems that 
, where 
but this baffles my intuition (which I’ll admit isn’t very sharp), especially for 
. Here 
is the Frobenius norm, and 
is the j-th column of 
.
Hopefully I’ll have time to look at this later. Besides being counterinuitive, it seems potentially useful: consider a vector of composite length 
, you could partition this vector into 
blocks of length 
then apply this inequality to get a family of bounds on the euclidean length of the vector. Exploit the fact that 
and you could derive some very weird bounds indeed.
Possibly relevant posts:
- One direction in Khintchine’s inequality for Rademacher sums (9/26/2008)
- Differentiable Flows (3/26/2005)
- But that’s just least squares… (8/9/2006)
For the operator norm, the $p=\infty$ case is the Schur Lemma.
But I doubt that it is true for Frobenius norm just the way you state it.
It isn’t; my officemate just came up with a counterexample:
Your norm is very asymmetric, so you could try to symmetrize it to throw counterexamples like that. But I still suspect that the inequality for the symmetrized version will not hold.
As in, add in the sum of the p-norms of the rows? I doubt that’d work also.