This does not look too complicated, but I've been stuck here for several hours. My question is to prove that $||(h, \cdots, h)||\leq ||h||^{k}$, where $||\cdot||$ is the euclidean norm, and $(h,\cdots, h)$ is a $k$-tuple with $h\in R^n$.
My attempt: by definition, $||(h, \cdots, h)||=|\sum\limits_{i_1, \cdots, i_k}^{n} (h_{i_1})\times \cdots \times (h_{i_k})|.$ Also, $||h||^{k}=((\sum\limits_{j=1}^{n} h_j^2)^k)^{\frac{1}{2}}.$ Then, I think I should use Cauchy-Swartz Inequality somewhere. Could someone give me a hint? Thanks in advance!
This is false; if $n=1$ and $k=2$ and $h=\frac{1}{2}$ then $||h||=\frac{1}{2}$ whereas $$\left|\left|\left(\frac{1}{2},\frac{1}{2}\right)\right|\right|=\sqrt{\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^2}=\frac{1}{\sqrt{2}}>\frac{1}{4}=||h||^2.$$ Of course this example scales to any $n$ and $k$ with $nk>1$.