Weak Convergence of a Sequence in a Real Inner Product Space

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, let $\mathbb{N}$ be the set of natural numbers, let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$, let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $E$, and let $x\in E$.

(i) We say that $(x_{m})_{m\in\mathbb{N}}$ converges weakly to $x$ in $E$, and write $x_{m}\rightharpoonup x$, if for every $y\in E$ the sequence of real numbers $(\langle x_{m},y\rangle)_{m\in\mathbb{N}}$ converges to $\langle x,y\rangle$. When several inner product spaces are in play we say that the sequence converges weakly in $E$, the inner product being that of $E$.