  For an $m$-order $n-$dimensional Hilbert tensor (hypermatrix)
$\mathcal{H}_n=(\mathcal{H}_{i_1i_2\cdots i_m})$, $$\mathcal{H}_{i_1i_2\cdots
i_m}=\frac1{i_1+i_2+\cdots+i_m-m+1},\ i_1,\cdots, i_m=1,2,\cdots,n$$ its
spectral radius is not larger than $n^{m-1}\sin\frac{\pi}{n}$, and an upper
bound of its $E$-spectral radius is $n^{\frac{m}2}\sin\frac{\pi}{n}$. Moreover,
its spectral radius is strictly increasing and its $E$-spectral radius is
nondecreasing with respect to the dimension $n$. When the order is even, both
infinite and finite dimensional Hilbert tensors are positive definite. We also
show that the $m$-order infinite dimensional Hilbert tensor (hypermatrix)
$\mathcal{H}_\infty=(\mathcal{H}_{i_1i_2\cdots i_m})$ defines a bounded and
positively $(m-1)$-homogeneous operator from $l^1$ into $l^p$ ($1<p<\infty$),
and the norm of corresponding positively homogeneous operator is smaller than
or equal to $\frac{\pi}{\sqrt6}$.
