fa.functional analysis - Radii and centers in Banach spaces

I believe that the property does not hold for all Banach spaces, but my counterexample is a little involved. If you've the patience then follow me through...

Let $V=bigoplus_{n=1}^infty ell^n_2$ where $ell^p_2$ is $mathbb{R}^2$ with
norm $lVertcdotrVert_p$ (Note: $n$ is taking the role of $p$). For $igeq1$
and $jin{0,1}$ we have $e_{i,j}$, the $j^{th}$ standard basis vector of
$ell^i_2$ in $V$.

Give $V$ the norm $lVert vrVert=sup_nlVert v_nrVert_n$.

Let $W={vin V:lVert v_nrVert_nto 0}$. I assert that $W$ is a Banach space.
Certainly every $e_{i,j}in W$.

Let $A={e_{k,0}+e_{k,1}, e_{k,0}-e_{k,1}:kgeq 1}$.

Fact: Let $r(A)$ be the infimum of radii of balls containing $A$. Then $r(A)leq1$

Proof:

Let $c_N=sum_{i=1}^n e_{i,0}$. We wish to compute the distance of each point
of $A$ from $c_N$.

For $kleq N$ we have
$lVert c_N-e_{k,0}-e_{k,1}rVert$
$=lVertsum_{i=1 (inot=k)}^Ne_{i,0}-e_{k,1}rVert$
$=sup{lVert e_{i,0}rVert_i:ileq N,inot=k}cup{lVert-e_{k,1}rVert_k}$
$=1$
and similarly for $lVert c_N-e_{k,0}+e_{k,1}rVert$.

For $k>N$ we have
$lVert c_N-e_{k,0}-e_{k,1}rVert$
$=max(lVert c_NrVert,lVert e_{k,0}+e_{k,1}rVert_k)$
$= max(1,(1+1)^frac{1}{k})$
$= 2^frac{1}{k}$
$leq 2^frac{1}{N}$
and similarly for $lVert c_N-e_{k,0}+e_{k,1}rVert$.

Thus $Asubseteq overline{B}(c_N,2^frac{1}{N})$ and so $r(A)leq2^frac{1}{N}$. Letting
$Ntoinfty$ we have $r(A)leq 1$.

QED

Fact: $A$ is not contained in a ball of radius $1$.

Proof:

Suppose $Asubseteq overline{B}(c,1)$. Then in particular for every $n$ we have
$lVert c-e_{n,0}-e_{n,1}rVertleq 1$ and thus $lVert c_n-e_{n,0}-e_{n,1}rVert_nleq 1$. Similarly $lVert c_n-e_{n,0}+e_{n,1}rVert_nleq 1$.

Simple consideration of $ell^n_2$ shows that this implies $c_n=e_{n,0}$.
Thus $lVert c_nrVert=1notto0$ and $cnotin W$, contradicting the assumption.

QED

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