Most elegant way to proof that the $\ell_1$-norm of a unit vector is larger equal the $\ell_2$-norm of it

Question

I was wondering which proof method can be used best to show that the $\ell_1$-norm of a unit vector is larger equal the $\ell_2$-norm of it?

Answer 1

More generally, this is true even without requiring that the vectors be unit, and for other $\ell_p$ norms: we have monotonicity of the $\ell_p$ norms: for any sequence $x=(x_1,\dots,x_n)\in\mathbb{R}^n$, $p > 0 \mapsto \lVert{x}\rVert_p$ is non-increasing.

In particular, for $0 < p \leq q <\infty$, $$ \left(\sum_i \lvert{x_i}\rvert^q\right)^{1/q} = \lVert{x}\rVert_q \leq \lVert{x}\rVert_p = \left(\sum_i \lvert{x_i}\rvert^p\right)^{1/p}\;. $$ To see why, one can easily prove that if $\lVert{x}\rVert_p = 1$, then $\lVert{x}\rVert_q^q \leq 1$ (bounding each term $\lvert{x_i}\rvert^q \leq \lvert{x_i}\rvert^p$), and therefore $\lVert{x}\rVert_q \leq 1 = \lVert{x}\rVert_p$. Next, for the general case, apply this to $y = x/\lVert{x}\rVert_p$, which has unit $\ell_p$ norm, and conclude by homogeneity of the norm.

Answer 2

Consider $$ \sum_{k=1}^n|x_n| = \left|\sum_{k=1}^n|x_n|\right| = \left(\left(\sum_{k=1}^n |x_n|\right)^2\right)^{1/2} \geq \left(\sum_{k=1}^n |x_n|^2\right)^{1/2} = \left(\sum_{k=1}^n x_n^2\right)^{1/2}, $$ because $y^2 = |y|^2$. If you expand the expression for $\left(\sum_{k=1}^n |x_n|\right)^2$ you get all of $\sum_{k=1}^n |x_n|^2$ plus some additional nonnegative terms. Also, the square root function $t\mapsto t^{1/2}$ is monotonic increasing.

Answer 3

Since you are dealing with a unit vector, $u$ each of the components $u_i$ must be at most one. Hence, for each $i$, $u_i^2 \le \lvert u_i\rvert$. So

$$ \lVert u\rVert_2^2 = \sum_i u_i^2 \le \sum_i \lvert u_i\rvert = \lVert u\rVert_1 $$