Ady, I think there is a counterexample to your question. To describe it, let (V_n)$(V_n)$ be a basis of [0,1]$[0,1]$ consisting of non-empy open sets; K$K$ stands for the closed unit ball of C[0,1]$C[0,1]$. For every n$n$ let C_n$C_n$ be the closure of V_n$V_n$ and define
U_n={g in K: min{|g(t)|:t in C_n} > |g| - 1/4}$U_n={g \in K: \min{|g(t)|:t \in C_n} > \|g\| - 1/4}$
where |g|=sup{|g(t)|:t in [0,1]}$\|g\|=\sup{|g(t)|:t \in [0,1]}$.
The family (U_n)$(U_n)$ is an open cover of K$K$. Let (F_m)$(F_m)$ be a partition of unity subordinate to (U_n)$(U_n)$. For every m let n_m$n_m$ be the least integer n$n$ such that supp(F_m)={g in K: F_m(g)>0}$\sup(F_m)={g \in K: F_m(g)>0}$ is contained in U_n$U_n$.
Now define F:K\to \mathbb{R}$F:K\to \mathbb{R}$ by
F(g)= \sum_{m=1}^{\infty} n_m\cdot F_m(g)$F(g)= \sum_{m=1}^{\infty} n_m\cdot F_m(g)$
Notice that F is well-defined and continuous.
Finally notice that F(K\cap E)$F(K\cap E)$ is unbounded for every infinite-dimensional subspace E of C[0,1]$C[0,1]$. This follows from the following fact: for every integer i and every infinte-dimensional subspace E$E$ of C[0,1]$C[0,1]$ there is a norm-one vector e in E$e \in E$ such e$e$ is NOT in U_n$U_n$ for every n < i$n < i$ (and therefore, if m$m$ is such that F_m(e)>0$F_m(e)>0$, then necessarily n_m$n_m$ is greater or equal to i$i$ which gives that F(e)$F(e)$ is also greater or equal to i$i$).
