If $\textbf{x} \in \mathbb{R}^n$, then show that the following inequalities hold. If $\textbf{x} \in \mathbb{R}^n$, then show that the following inequalities hold
$||\textbf{x}||_\infty \leq ||\textbf{x}||_2 \leq \sqrt{n}||\textbf{x}||_\infty$
$\frac{1}{\sqrt{n}}||\textbf{x}||_1 \leq ||\textbf{x}||_2 \leq \sqrt{n}||\textbf{x}||_1$

I have been trying to read about $L^\infty$ spaces and not sure if this translates to vector norms. I read that a vector norm is the max entry value. I would like some help on a definition of the terms shown above and how to being this problem. I have tried to think about how an infinite vector could possibly have an entry with larger value than a vector in $\mathbb{R}^2$. I am not sure if I am even on the right track in thinking about this problem.
 A: If you are looking for definitions, in the finite dimensional space $\mathbb R^n$, if $\mathbf x \in \mathbb R^n$ with coordinates $x_i$, for $1 \leqslant i \leqslant n$
\begin{align}
\lVert \mathbf x \rVert_\infty &= \max_{1 \leqslant i \leqslant n} \lvert x_i \rvert \\ 
\lVert \mathbf x \rVert_2 &= \sqrt {\sum_{i=1}^n |x_i|^2 }\\
\lVert \mathbf x \rVert_1 &= \sum_{i=1}^n |x_i|.
\end{align}
The inequalities you state then follow pretty directly from these definitions.
The concepts also extend to infinite dimensions.  There are three vector spaces that correspond to those above.
The first, written $\ell_\infty$ is the space of all bounded sequences $x_i, 1 \leqslant i$ in which the norm is defined as $\lVert \mathbf x \rVert_\infty = \sup_{1 \leqslant i} | x_i |$.  The supremum exists because the sequence must be bounded to be contained in the space $\ell_\infty$.
A few examples:  the sequence $(x_i) = (1,2,3,4,\cdots)$ is not in $\ell_\infty$ because it is unbounded.  The two sequences $(1,1,1,\cdots)$ and $(1,1/2,1/3,\cdots)$ are both in $\ell_\infty$ because they are bounded.  Both have $\ell_\infty$ norm of $1$.
The second space is $\ell_2$ with corresponding norm $\sqrt{\sum_{i=1}^\infty |x_i|^2}$, where members of the space are required to have finite norm.  Of the three sequences above, only the last is in $\ell_2$ with norm $\pi/\sqrt 6$,as the norms of the other two would be infinite, thereby disqualifying the sequences from inclusion in the space $\ell_2$.
The third, $\ell_1$ has norm $\sum_{i=1}^\infty |x_i|$, and none of the three examples are in the space because all there have unbounded norms.
