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Let $H$ be infinite-dimensional Hilbert space with basis functions $b_i$.

Let $B_n = \text{span}\{b_1, ...,b_n\}$.

So $\text{dim}(N) = n$.

Let $c_i$ be another basis for $H$. Is it true that $$B_n = \text{span}\{c_{j_1}, ..., c_{j_n}\}$$ for some indices ${j_i}$?

I think so since $B_n$ has dimension $n$ so anything in the set can be written as sum of $n$ basis functions?

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It is not even true for finite-dimensional spaces. Let $B = \mathrm{span}(e_1) \subseteq \mathbb{R}^2$ and look at the basis $c_1 = (1,1)$, $c_2 = (1,-1)$ of $\mathbb{R}^2$.

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