Elementary Properties of Series in a Real Inner Product Space

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $E$ and $F$ be real inner product spaces, let $H$ be a real Hilbert space, let $(x_{k})_{k\in\mathbb{N}}$ and $(y_{k})_{k\in\mathbb{N}}$ be sequences in $E$ with partial sums $s_{n}$ and $t_{n}$ respectively, let $\lambda\in\mathbb{R}$, let $w\in E$ and let $T\in\mathcal{L}(E,F)$. Convergence of a series in a real inner product space, its sum, and absolute convergence are as defined there, and series of real numbers are as in Series of Real Numbers. Then the following hold.

1. (Linearity)(i) If $\sum_{k=1}^{\infty}x_{k}$ and $\sum_{k=1}^{\infty}y_{k}$ converge, then $\sum_{k=1}^{\infty}(x_{k}+y_{k})$ and $\sum_{k=1}^{\infty}(\lambda x_{k})$ converge, and

$$\sum_{k=1}^{\infty}(x_{k}+y_{k})=\sum_{k=1}^{\infty}x_{k}+\sum_{k=1}^{\infty}y_{k}, \qquad \sum_{k=1}^{\infty}(\lambda x_{k})=\lambda\sum_{k=1}^{\infty}x_{k}.$$

2. (The terms tend to zero)(ii) If $\sum_{k=1}^{\infty}x_{k}$ converges, then $(x_{k})_{k\in\mathbb{N}}$ converges to the zero vector $0_{E}$.

3. (Telescoping series)(iii) Let $(z_{k})_{k\in\mathbb{N}}$ be a sequence in $E$ with $x_{k}=z_{k+1}-z_{k}$ for every $k\in\mathbb{N}$. Then $s_{n}=z_{n+1}-z_{1}$ for every $n\in\mathbb{N}$; consequently $\sum_{k=1}^{\infty}x_{k}$ converges if and only if $(z_{k})_{k\in\mathbb{N}}$ converges, and in that case its sum is $\bigl(\lim_{k\to\infty}z_{k}\bigr)-z_{1}$.

4. (Cauchy criterion)(iv) Let $(x_{k})_{k\in\mathbb{N}}$ be a sequence in the real Hilbert space $H$, with partial sums $s_{n}$. Then $\sum_{k=1}^{\infty}x_{k}$ converges if and only if for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ such that $|s_{n}-s_{m}|<\varepsilon$ for all $m,n\in\mathbb{N}$ with $N\le m$ and $N\le n$.

5. (Absolute convergence)(v) Let $(x_{k})_{k\in\mathbb{N}}$ be a sequence in the real Hilbert space $H$ such that $\sum_{k=1}^{\infty}x_{k}$ converges absolutely. Then $\sum_{k=1}^{\infty}x_{k}$ converges and

$$\Bigl|\sum_{k=1}^{\infty}x_{k}\Bigr|\le\sum_{k=1}^{\infty}|x_{k}|.$$

6. (Bounded linear maps)(vi) If $\sum_{k=1}^{\infty}x_{k}$ converges, then the series $\sum_{k=1}^{\infty}Tx_{k}$ converges in $F$ and

$$\sum_{k=1}^{\infty}Tx_{k}=T\Bigl(\sum_{k=1}^{\infty}x_{k}\Bigr).$$

7. (Inner products against a series)(vii) If $\sum_{k=1}^{\infty}x_{k}$ converges, then the series of real numbers $\sum_{k=1}^{\infty}\langle x_{k},w\rangle$ converges and

$$\sum_{k=1}^{\infty}\langle x_{k},w\rangle=\Bigl\langle\sum_{k=1}^{\infty}x_{k},\,w\Bigr\rangle .$$

8. (Agreement on the real line)(viii) Let $E$ be the real inner product space $\mathbb{R}$ of Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces (vii) and let $(a_{k})_{k\in\mathbb{N}}$ be a sequence of real numbers. Then the partial sums of $(a_{k})$ in the sense of this item and in the sense of that item coincide, and the series $\sum_{k=1}^{\infty}a_{k}$ converges in one sense if and only if it converges in the other, with the same sum.