Find the vector of minimum norm... I'm working on this exercise:

Show that the subset $ M = ${$y= (\eta_j) ;  \sum \eta _j= 1$} of
complex space $\mathbb{C^{n}}$  is complete and convex. Find the
vector of minimum norm in M.

I've proved the completeness and convexity part, but I could not find the minimizing vector. I have the suspicion that it's $(\frac{1}{n},\frac{1}{n},...,\frac{1}{n})$, but I couldn't find an argument.
Could you help me? Thank you in advance.

This also follows directly from the Cauchy-Schwarz inequality:
$$
\sum_{j=1}^n \eta_j=\sum_{j=1}^n \eta_j\cdot 1\leq \left(\sum_{j=1}^n |\eta_j|^2\right)^{1/2}\left(\sum_{j=1}^n 1\right)^{1/2}=n^{1/2}\left(\sum_{j=1}^n |\eta_j|^2\right)^{1/2}
$$
With equality if and only if there exists $\lambda\geq 0$ such that $\eta_j=\lambda$ for all $j\in\{1,\dots,n\}$. The constraint $\sum_j \eta_j=1$ is then satisfied if and only if $\lambda=1/n$.
