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We study the metric projection onto the closed convex cone in a real Hilbert space 
$\mathscr {H}$
generated by a sequence 
$\mathcal {V} = \{v_n\}_{n=0}^\infty $
. The first main result of this article provides a sufficient condition under which the closed convex cone generated by 
$\mathcal {V}$
coincides with the following set:
$$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$
Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto 
$\mathcal {C}[[\mathcal {V}]]$
. As an application, we obtain the best approximations of many concrete functions in 
$L^2([-1,1])$
by polynomials with nonnegative coefficients.
