Equivalence norms property

Question

If $||.||_1$ and $||.||_2$ are two equivalent norms such that $$||a||_1 \le ||b||_1$$ then do we have $$||a||_2 \le ||b||_2$$ ?

Answer 1

Not at all for example in $\mathbb R^2$ take $a = (1,0)$ $b = \left(\frac12, \frac12\right)$. You have $\|a\|_1 = 1 = \|b\|_1$ but $\|a\|_2 = 1 > \frac1{\sqrt{2}} = \|b\|_2$