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Let $H$ be a Hilbert space, consisting of functions $f:\mathbb{R} \to \mathbb{R}$. Let $$ V = \left\{ f_J \in H: f_J= \sum_{j=1}^J c_j^{(J)} g_j, c_j^{(J)}\in\mathbb R, J\in \mathbb N \right\} $$ where $g_j \in H$ for each $j\in \mathbb N$.

My question: If provided that $V$ is dense in $(H,\|\cdot\|_H)$, does any $f\in H$ admit a representation as follows? $$ f = \sum_{j=1}^\infty c_j g_j $$ If not, what type of additional assumptions will be needed?

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    $\begingroup$ No. Polynomials are dense in $L^2(0,1)$. $\endgroup$
    – Christian Remling
    Commented Jan 29 at 22:32
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    $\begingroup$ and Gram orthonormalization of the sequence g_j produces a sequence h_j that makes it possible $\endgroup$
    – Pietro Majer
    Commented Jan 30 at 7:16

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