I'm trying to figure out if there is a vector $x ∈ R^{d} $ such that $||x||_{1}$= 100 and $||x||_{∞} = 1$. I need to either give an example of a vector or prove that no such vector exists.
I know that the $L^{1}$ norm of a vector x = $(x_{1},...,x_{d}) ∈ R^{d}$ is just $|x_{1}| + . . . + |x_{d}|$, and the $L^{∞}$ norm is the largest of the numbers. I'm not too sure where to start other than trying random vectors that only hold true for $L^{1}$ but not $L^{∞}$
It is clear from the defintions of the norms that $\|x\|_1 \leq d\|x\|_{\infty}$. So we cannot have such an $x$ if $100> d$. If $100 \leq d$ look at a vector of the form $(1,1,...,1,0,0,...)$.