Weighted Square-Summable Sequence Spaces are Real Hilbert Spaces: the Standard Orthonormal Basis and the Embedding of a Hilbert Space under Bounded Weights

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinates $x_{k}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, let $a=(a_{k})_{k\in\mathbb{N}}$ be a sequence of positive real numbers, let $X_{a}$ with its termwise operations be the weighted sequence space, and let $\langle\cdot,\cdot\rangle_{a}$ and $|\cdot|_{a}$ be as in Weighted Square-Summable Sequence Spaces §inner-product.

1. (A real Hilbert space)¶ With its termwise operations and $\langle\cdot,\cdot\rangle_{a}$, $X_{a}$ is a real Hilbert space whose zero vector is the zero sequence, and $|\cdot|_{a}$ is the norm of its inner product.

2. (Standard orthonormal basis)¶ For $k\in\mathbb{N}$ let $f^{a}_{k}$ be the sequence whose $k$-th term is the reciprocal of $a_{k}^{1/2}$ and whose other terms are $0$. Then $f^{a}_{k}\in X_{a}$, $(f^{a}_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $X_{a}$, and $\langle v,f^{a}_{k}\rangle_{a}=a_{k}^{1/2}v_{k}$ for every $v\in X_{a}$ and $k\in\mathbb{N}$.

3. (Embedding under bounded weights)¶ Suppose that there is $\bar{a}\in\mathbb{R}$ with $a_{k}\le\bar{a}$ for every $k\in\mathbb{N}$. Then for every $x\in X$ the sequence $\iota_{a}(x)=(x_{k})_{k\in\mathbb{N}}$ belongs to $X_{a}$, the map $\iota_{a}:X\to X_{a}$ is linear for the termwise operations and injective, and $|\iota_{a}(x)|_{a}^{2}\le\bar{a}\,|x|^{2}$ for every $x\in X$.