The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights

In the settings of Real Hilbert Spaces: Standing Notation and Background and Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus, let $H$ be a real Hilbert space, written in the ambient notation $\langle\cdot,\cdot\rangle_{H}$, $|\cdot|_{H}$, $d_{H}$ and with zero vector $0_{H}$, let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$, and let $(\lambda_{k})_{k\in\mathbb{N}}$ be a sequence of real numbers with $1\le\lambda_{k}$ for every $k\in\mathbb{N}$. For $x\in H$ write $x_{k}=\langle x,e_{k}\rangle_{H}$.

Let $V$ be the set of those $x\in H$ for which the series $\sum_{k=1}^{\infty}\lambda_{k}x_{k}^{2}$ converges. By claim 1 below, for $x,y\in V$ the series $\sum_{k=1}^{\infty}\lambda_{k}x_{k}y_{k}$ converges; its sum is written $\langle x,y\rangle_{V}$, and $|\cdot|_{V}$ and $d_{V}$ denote the associated norm and distance. Then the following hold.

1. (A subspace carrying an inner product)(i) The set $V$ is a linear subspace of $H$ and contains $e_{j}$ for every $j\in\mathbb{N}$. For all $x,y\in V$ the series $\sum_{k=1}^{\infty}\lambda_{k}|x_{k}y_{k}|$ and $\sum_{k=1}^{\infty}\lambda_{k}x_{k}y_{k}$ converge, and $\langle\cdot,\cdot\rangle_{V}$ makes $V$ a real inner product space whose norm satisfies

$$|x|_{H}\le|x|_{V}\qquad\text{for every }x\in V .$$

2. (Coefficients against the basis)(ii) For every $x\in V$ and every $j\in\mathbb{N}$,

$$\langle x,e_{j}\rangle_{V}=\lambda_{j}x_{j}, \qquad\text{and in particular}\qquad |e_{j}|_{V}^{2}=\lambda_{j}.$$

3. (Completeness)(iii) The space $V$ with $\langle\cdot,\cdot\rangle_{V}$ is a real Hilbert space.

4. (Density)(iv) The set $V$ is dense in $H$.

5. (A rescaled basis, and separability)(v) For $k\in\mathbb{N}$ let $\mu_{k}$ be the nonnegative real number with $\mu_{k}^{2}=\tfrac{1}{\lambda_{k}}$ given by Existence and Uniqueness of the Nonnegative Square Root. Then $\mu_{k}$ is positive, the sequence $(\mu_{k}e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $V$, and the metric space $(V,d_{V})$ is separable.