91Ó°ÊÓ

Prove that the elements of a convergent sequence in a normed linear space always form a bounded set.

Let \(X\) be a normed linear space, and let \(f\) be a continuous function from \(X\) to \(\mathbb{R}\). Show that the function \(|f|\) defined by \(|f|(x)=|f(x)|\) for \(x \in X\) is continuous.

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be continuous on the interval \([a, b]\). Define \(g:[a, b] \rightarrow \mathbb{R}\) by \(g(s)=\max \\{f(t): t \in[a, s]\\}\) if \(s \neq a\) and \(g(a)=f(a)\). Show that \(g\) is continuous on \([a, b]\). We shall conclude this section by examining the continuity of functions. on the linear space \(\mathbb{R}^{n}\). Recall that \(\mathbf{R}^{n}\) is the set of \(\mathrm{n}\)-tuples of the form $$ \left(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right) $$ where each entry \(x_{i}, 1 \leq i \leq n\), is a real number. The usual definitions of addition and multiplication by scalars are adopted, so that addition of two n-tuples is effected by adding the individual components and multiplication by a scalar simply means that each member of the ntuple is multiplied by that scalar. Hence, $$ \left(x_{1}, x_{2}, \ldots, x_{n}\right)+\left(y_{1}, y_{2}, \ldots, y_{n}\right)=\left(x_{1}+y_{1}, x_{2}+y_{2}, \ldots, x_{n}+y_{n}\right) $$ and $$ \alpha\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\left(\alpha x_{1}, \alpha x_{2}, \ldots, \alpha x_{n}\right) $$ There are three principal definitions which we adopt for the norm in \(\mathbb{R}^{n}\). If we assume that \(x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) then these norms are defined as follows: $$ \|x\|_{2}=\left\\{\sum_{i=1}^{n} x_{i}^{2}\right\\}^{\frac{1}{2}}, \quad\|x\|_{1}=\sum_{i=1}^{n}\left|x_{i}\right|, \quad\|x\|_{\infty}=\max \left\\{\left|x_{i}\right|: 1 \leq i \leq n\right\\} $$ It is perhaps a little surprising that the verification that the first of these definitions really has the properties required of a norm is quite difficult. The other two are straightforward, but one of the exercises guides you through the proof of the first case. There are sensible historical reasons for the subscripting of these norms, which come from a much larger family of norms on \(\mathbb{R}^{n}\). Throughout this section the symbol \(\|\cdot\|\) is used to denote any one of the three norms above and unless mention is made of a particular norm any statement about convergence or continuity should be understood to refer to each of the three norms. The reason for this ambiguity is contained in the next lemma

A function \(f: \mathbb{R}^{n} \rightarrow \mathbb{R}\) is said to be additive if $$ f(x+y)=f(x)+f(y), \quad \text { for all } x, y \text { in } \mathbb{R}^{n} $$ Show that an additive function which is continuous at \(x=\theta\) is continuous everywhere in \(\mathbb{R}^{n}\).