Let $\mathscr{H}$ be a Hilbert space and suppose $f$ and $g$ are linearly independent vectors in $\mathscr{H}$ with $\|f\|=\|g\|=1$. Show that $\|tf + (1-t)g\| < 1$ for all $0 < t < 1$. What does this say about $\{h \in \mathscr{H} \colon \|h\| \leq 1\}$.
I have proven the statement, but I am unsure about what there is to say about the subset $\{h \in \mathscr{H} \colon \|h\| \leq 1 \}$. Clearly it is a convex subset of $\mathscr{H}$, but what else can we say about it that is of interest?
It is not only convex, but strictly convex! (this is an important property in the theory of normed vector spaces)