Example

\* In Hilbert space $H$ consisting of sequences $x = (x_n)$ of real numbers such that

the neighbourhoods of a point $x_0$ in the strong topology on $H$ are the sets which contain an open ball $\|x - x_0\| < \alpha$ centred at $x_0$; the neighbourhoods of $x_0$ in the weak topology on $H$ are the sets containing a set defined by a relation of the form $\sup_{1 \leq i \leq n} |(x - x_0|a_i)| \leq 1$, where the $a_i$ are points of $H$ and if $x = (x_n)$ and $y = (y_n)$. Now if $\beta = \sup_{1 \leq i \leq n} \|a_i\|$, the relation $\|x - x_0\| \leq 1 / \beta$ implies that $|(x - x_0|a_i)| \leq \|x - x_0\| \cdot \|a_i\| \leq 1$ for $1 \leq i \leq n$; hence the strong topology on $H$ is finer than the weak topology. On the other hand, given any finite family $(a_i)_{1 \leq i \leq n}$ of points of $H$, there are points $x$ in $H$ such that $(x - x_0|a_i) = 0$ for $1 \leq i \leq n$ and such that $\|x - x_0\|$ is arbitrarily large; this shows that the strong topology is strictly finer than the weak topology. \*