Elementary Properties of Weak Convergence in a Real Inner Product Space

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, and for $s\in\mathbb{R}$ let $|s|$ be its absolute value. Let $\mathbb{N}$ be the set of natural numbers, let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$ (the symbol $|\cdot|$ denoting the norm or the absolute value according to the type of its argument), let $(x_{m})_{m\in\mathbb{N}}$ and $(y_{m})_{m\in\mathbb{N}}$ be sequences in $E$, let $x,y\in E$, let $\lambda\in\mathbb{R}$, and let $C\in\mathbb{R}$. Weak convergence is written $\rightharpoonup$, convergence of sequences in $E$ is convergence in $(E,d)$, and convergence of real sequences is as in Limit of a Sequence of Real Numbers. Then the following hold.

1. (Uniqueness)(i) If $x_{m}\rightharpoonup x$ and $x_{m}\rightharpoonup y$, then $x=y$.

2. (Strong implies weak)(ii) If $(x_{m})$ converges to $x$ in $(E,d)$, then $x_{m}\rightharpoonup x$.

3. (Linearity)(iii) If $x_{m}\rightharpoonup x$ and $y_{m}\rightharpoonup y$, then $x_{m}+y_{m}\rightharpoonup x+y$, $x_{m}-y_{m}\rightharpoonup x-y$ and $\lambda x_{m}\rightharpoonup\lambda x$.

4. (Subsequences)(iv) If $x_{m}\rightharpoonup x$, then every subsequence of $(x_{m})$ converges weakly to $x$.

5. (Norm bound passes to the limit)(v) If $x_{m}\rightharpoonup x$ and $|x_{m}|\le C$ for every $m\in\mathbb{N}$, then $|x|\le C$.

6. (Radon-Riesz)(vi) If $x_{m}\rightharpoonup x$ and the real sequence $(|x_{m}|)$ converges to $|x|$, then $(x_{m})$ converges to $x$ in $(E,d)$.

7. (Pairing)(vii) If $x_{m}\rightharpoonup x$, $|x_{m}|\le C$ for every $m\in\mathbb{N}$, and $(y_{m})$ converges to $y$ in $(E,d)$, then the real sequence $(\langle x_{m},y_{m}\rangle)$ converges to $\langle x,y\rangle$.

8. (Dense criterion)(viii) Let $D\subseteq E$ be dense in $E$. If $|x_{m}|\le C$ for every $m\in\mathbb{N}$ and $(\langle x_{m},z\rangle)$ converges to $\langle x,z\rangle$ for every $z\in D$, then $x_{m}\rightharpoonup x$.