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In her book "Spline models for observational data", on page 4, Grace Wahba states that the following function space is "obviously" complete:

Let $\{\phi_{i}\}_{i=1..\infty}$ be a sequence of continuous functions which is orthonormal in $L^{2}$ (i.e. $\phi_{i}$ is an ONB), and $\{\lambda_{i}\}_{i=1..\infty}$ a sequence of real numbers. I dont know if it is necessary to know, but $\phi_{i},\lambda_{i}$ are the (Mercer) eigenvectors/eigenvalues belonging to an integral operator with a positive definite kernel $K$, and it holds that \begin{align} \int K(x,t) \phi_{i}(t) dt = \lambda_{i}\phi_{i}(x) \\ K(x,y) = \sum_{i=1}^{\infty} \lambda_{i}\phi_{i}(x) \phi_{i}(y) \\ \sum_{i=1}^{\infty} \lambda_{i}^{2} < \infty \end{align}

In the book it is then stated that the subspace of $L^{2}$ given by \begin{equation} \{f \mid f=\sum_{i=1}^{\infty}\alpha_{i} \phi_{i}, \sum_{i=1}^{\infty}\frac{\alpha_{i}^{2}}{\lambda_{i}}<\infty \} \end{equation} , equipped with the norm $$\Vert f \Vert = \sum_{i=1}^{\infty}\frac{\alpha_{i}^{2}}{\lambda_{i}} $$ is "obviously" complete.

I have tried to show this, without luck. Could anyone explain?

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    I haveo only one question. $\lambda_{i}$-s is sequence of arbitrary reals? Or it is the sequence for which the ordinary norm is exactly as written? – 2017-01-20
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    I added some more information about the context in which it is presented in the book. I dont know if it is necessary for the completeness to hold though. – 2017-01-20
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    One must assume $\lambda_i > 0$ for all $i$, or take the absolute value of $\lambda_i$ (and one must still assume $\lambda_i \neq 0$ for all $i$ then). Assuming $\lambda_i > 0$, the sequence space $$\Biggl\{ \alpha \colon \mathbb{N}\to \mathbb{R} : \sum_{i = 1}^{\infty} \frac{\alpha_i^2}{\lambda_i} < +\infty\Biggr\}$$ is just $L^2(\mu)$, where $\mu$ is the measure given by $$\mu(A) = \sum_{i \in A} \frac{1}{\lambda_i}.$$ $L^2(\mu)$ is complete. Now just check that $\alpha \mapsto \sum \alpha_i\phi_i$ is an isometry between $L^2(\mu)$ and your space. – 2017-01-20
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    I forgot one important question: can we be sure that the series $\sum^{\infty}_{i=1} \alpha_{i} \phi_{i}$ converges, given the assumptions above? – 2017-04-25

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