= Solution
\b[No.] Take the <Hilbert space> $H=\ell^2$ and its standard <orthonormal sequence> $(e_n)$. Let
$$
C=\{u\in\ell^2:\lVert u\rVert_2=1\},
$$
the unit sphere. This set is norm closed but is not convex. By the <Bessel inequality>, for every $y=(y_j)\in\ell^2$,
$$
\langle e_n,y\rangle=\overline{y_n}\longrightarrow0,
$$
so $e_n\rightharpoonup0$; equivalently, this is a <weakly null orthonormal sequence>. Every $e_n$ lies in $C$, but the weak limit $0$ does not. Hence convexity is essential.
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