Let $V$ be a vector space over a field $F$ and let $f$ be a function from $V$ to the interval $I:=[0,1]$ satisfying the condition that for any $a \in I$ the set $V_a:=\{v \in V | f(v) \ge a\}$ is a subspace of $V$.
Then I believe that there exists a basis $(\alpha_i)_{i\in \Omega}$ satisfying the following:
By definition each vector $b \in V$ is representable as a linear combination of some vectors of that basis, for example, $b = t_1 \alpha_{k_1} + t_2 \alpha_{k_2} + ... + t_j \alpha_{k_j} $ where $t_i$ are nonzero. Then we would have $f(b) = \min \{f(\alpha_{k_1}), ... , f(\alpha_{k_j}) \}$.
May you help me to prove (or disprove!) that statement? Thank you. More info is in the comment.