The reduced SVD of $B_{m\times n}$ is $USV^T$ and we know the columns of $U$ and $V$ are orthonormal. $e_i$ is the $i$th standard basis vector. I want to know the range of the following optimization problem.
$u_1=max_{i=1,...,m} ||UU^Te_i||^2$
What I have done is: expend $||UU^Te_i||^2=e_i^TUU^TUU^Te_i$, because the columns of $U$ is orthonormal, so $U^TU=I$. Then $||UU^Te_i||^2=e_i^TUU^Te_i=(U^Te_i)^TU^Te_i = ||U^Te_i||^2$. Therefore, the above optimization problem becomes
$u_1=max_{i=1,...,m} ||UU^Te_i||^2=max_{i=1,...,m}||U^Te_i||^2$
Someone claims that $\frac{1}{m}\leq u_1 \leq 1$, but I don't see it because we don't know about the property the rows of $U$. How can we find the range? Please help me:)