Remark 1

If $\sum_{n=0}^{\infty} \|u_n\| < +\infty$, then $\sum_{n=0}^{\infty} |u_n(x)| \leq \sum_{n=0}^{\infty} |u_n| < +\infty$ for each $x \in X$; in other words, for each $x \in X$ the series with general term $u_n(x)$ is absolutely convergent in the space $Y$. The converse is false. To avoid all confusion we shall sometimes say that the series with general term $u_n$ is normally convergent, meaning that the series with general term $\|u_n\|$ is convergent. A series can be uniformly convergent in $X$ without being normally convergent; this is the case, for example, for the series $(u_n)$ in the space $\mathscr{B}(\mathbf{R}, \mathbf{R})$, defined as follows: $u_n(x) = (1/n) \sin x$ if $x \in [n\pi, (n+1)\pi]$, $u_n(x) = 0$ otherwise.

When $Y$ is a normed algebra (Chapter IX, § 3, no. 7) over a non-discrete valued field $K$, then $\mathscr{B}(X; Y)$ is a $K$-algebra, and the norm $\|u\|$ is compatible with the algebra structure, since

Thus $\mathscr{B}(X; Y)$ is now a normed algebra over $K$.