Weighted Square-Summable Sequence Spaces

In the setting of The Real Numbers: Standing Notation and Background, with convergence of a series of real numbers and its sum as defined there, let $a=(a_{k})_{k\in\mathbb{N}}$ be a sequence of positive real numbers.

1. (The space)¶ The weighted sequence space $X_{a}$ is the set of the sequences $v=(v_{k})_{k\in\mathbb{N}}$ of real numbers for which the series $\sum_{k=1}^{\infty}a_{k}v_{k}^{2}$ converges. Sums and real multiples of elements of $X_{a}$ are taken termwise; they belong to $X_{a}$ by Products and Sums of Weighted Square-Summable Sequences of Real Numbers §sums, applied with the weights $\mu_{k}=a_{k}$ written there and the two sequences in place of $(a_{k})$ and $(b_{k})$ there.

2. (Inner product and norm)¶ For $v,w\in X_{a}$,

$$\langle v,w\rangle_{a}=\sum_{k=1}^{\infty}a_{k}v_{k}w_{k},\qquad|v|_{a}=\sqrt{\langle v,v\rangle_{a}}\ ,$$

the series converging by Products and Sums of Weighted Square-Summable Sequences of Real Numbers §products, applied in the same way; $\langle v,v\rangle_{a}=\sum_{k=1}^{\infty}a_{k}v_{k}^{2}$ is nonnegative by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, its terms being nonnegative.