fa.functional analysis - Selecting basic sequences

Not a basic sequence. Consider $e_0 oplus e_gamma$ in $Roplus H$ for $H$ a non separable Hilbert space, or, if you want a separable example, make $e_gamma$ a Hamel basis for a separable Hilbert space.

EDIT: Aug 2. Every separated sequence of unit vectors contains a minimal subsequence with bounded biorthogonal functionals. There remains the case where your linearly independent set has compact closure. I have an example of such a set where any minimal sequence in the set has only unbounded biorthogonal functionals, but I do not know whether such a set must contain a minimal sequence.

Too bad you did not ask this a day earlier when we could have discussed it face to face. I'll write something down when I get a chance.

EDIT: Aug 2. It was not so bad to write--I was able to do it on the plane.

If $x_n$ is a separated subset of the unit sphere of $X$, then $x_n$ has a minimal subsequence whose biorthogonal functionals are uniformly bounded. Indeed, if $x_n$ does not have weakly compact closure, then it has a basic subsequence (see e.g. the book of Albiac-Kalton), so we can assume that $x_nto x$ weakly. If $x=0$, then $x_n$ has a basic subsequence. If not, let $Q$ be the quotient map from $X$ onto $X/[x]$, where $[x]$ is the linear span of $x$. $Qx_nto 0 $ weakly and is bounded away from zero by the separation assumption, hence has a basic subsequence $Qx_{n(k)}$, whence $x_{n(k)}$ is minimal with uniformly bounded biorthogonal functionals.

I don't know what can happen when $A$ is a linearly independent subset of the unit sphere of $X$ that is totally bounded, but there exists such sets so that every minimal sequence in the set has biorthogonal functionals that are not uniformly bounded. Consider the Cantor set as the branches of the infinite binary tree and let the nodes index the unit vector basis of $c_0$. Given a branch $t=(t(n))$ (where $t_1<t_2<dots$ in the tree ordering) of the tree, let $x_t=sum 2^{-n} e_{t(n)}$. By compactness (which is really just pigeonholing), any sequence $y_k$ of $x_t$-s has a subsequence that converges to some $x_s$, which means that for any $n$, if $kge k(n)$ then $y_k(j)=x_s(j)$ for $1le j le n$. From this it is easy to see that $y_k$ cannot be uniformly minimal.

In the above argument, try replacing the unit vector basis of $c_0$ with an appropriate normalized countably linearly independent sequence that has no minimal subsequence. I think it is known that such sequences exist. Probably an example is in Kadec's book. Maybe this will give an example that has no infinite minimal subset.