Bound on l1 norm given bound on l2 norm

Question

While doing self-study exercices, I found the following bound without explanation and was not able to see why it is always the case. I found some examples, it seems legitimate but I am unable to produce a formal proof of the point. Here is the property:

Given an arbitrary unit vector (wrt $\ell_2$) $v$ in $\mathbb{R}^d$, then $||v||_1 \leq \sqrt{d}$

Answer 1

Since $v$ is unitary $v_i^2 \leq 1$. Using Jensen's inequality with $\phi(x)=x^2$: $$\lVert v \rVert_1^2 = \phi\Bigl(\sum_{i=1}^d \lvert v_i \rvert \Bigr) \leq \sum_{i=1}^d \phi(v_i) \leq d$$

Answer 2

Apply Cauchy-Schwarz:

$$\|v \|_1 = \sum_{i=1}^d |v_i| = \sum_{i=1}^d |v_i * 1| \le \left(\sum_{i=1}^d |v_i|^2\right) \left(\sum_{i=1}^d 1^2\right) =\|v\|_2 \sqrt d $$