Must the image of unit ball be bounded?

Question

I am going through a series of true or false questions, one of them is:

If $\phi:U\to V$ is a linear, $U,V$ are normed vector spaces, must $\text{im}\,\phi\big|_S$ be bounded, where$S=\{x\in U:\;||x||\leqslant1\}?$

It feels to me like the constraints are not strong enough for the answer to be yes, so I tried to find a counterexample:

Let $U=\ell^{\infty}$ and $V=\ell^1$ and $\phi$ be the identity map. Define $x^{(n)}=(1,1,\dots,1, 0,0\dots)$. Then in $U$, $x^{(n)}\in S$ for all $n$ and converges since constant, but in $V$, $||x^{(n)}||_1\to\infty$ so cannot be bounded.

Does this work? Seemed a little simple but it would be nice if it did.

Answer 1

Let $U\subset\ell^\infty$ be the subspace of sequences $x=(x_k)_{k\geq1}\in\ell^\infty$ with compact support, and define $\phi:\>U\to{\mathbb R}$ by $$\phi(x):=\sum_{k=1}^\infty x_k\ .$$