Let $C([0,1])$ denote the vector space of continuous functions $u : [0,1] \rightarrow \mathbb{R}$. On $C([0,1])$, consider the norm defined by
$$\left\Vert \varphi \right\Vert = \sup _{t \in [0,1]} \left| \varphi(t) \right| \quad (\forall \varphi \in C([0,1]).$$
Set
$$C := \bigg \{ u \in C([0,1]) : \int_0^1 \left| u(t) \right| ^2 dt < 1 \bigg \}$$
Show that $C$ is an open and convex subset of $C([0,1]).$
Let $I$ be an interval in $\mathbb{R}.$
For $k \in \mathbb{N}$, let $g_k, h_k \in L^2 (I)$ where
$$\sum_{k \in \mathbb{N}} \left\Vert g_k \right\Vert_2^2 < \infty \quad \text{and} \quad \sum_{k \in \mathbb{N}} \left\Vert h_k \right\Vert_2^2$$
and
$$\left\Vert g \right \Vert_2 = \bigg [\int_{I...