The Diagonal Hilbert Triple 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 $\langle\cdot,\cdot\rangle_{H}$, $|\cdot|_{H}$, $d_{H}$, 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 $k\in\mathbb{N}$. Let $V$ with its inner product $\langle\cdot,\cdot\rangle_{V}$ be the weighted coefficient subspace these data determine, as in The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights.

(i) The diagonal Hilbert triple determined by $(e_{k})_{k\in\mathbb{N}}$ and $(\lambda_{k})_{k\in\mathbb{N}}$ is the Hilbert triple $(H,V,A)$ formed by $H$, by $V$ with the inner product $\langle\cdot,\cdot\rangle_{V}$, and by the form operator $A$ that these two spaces determine. The numbers $\lambda_{k}$ are called the weights of the triple.

The requirements that definition makes of the pair $H$, $V$ are met: $V$ is a linear subspace of $H$ and satisfies the embedding inequality $|x|_{H}\le|x|_{V}$ for $x\in V$ by The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (i); $V$ with $\langle\cdot,\cdot\rangle_{V}$ is a real Hilbert space by The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (iii); and $V$ is dense in $H$ by The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (iv).