Essentially the only way in which we exploited, so far, the existence of an inner product in an inner product space was to introduce the notion of a normal transformation together with certain important special cases of it. A much more obvious circle of ideas is the study of the convergence problems that arise in an inner product space.

Let us see what we might mean by the assertion that a sequence \((x_{n})\) of vectors in \(\mathcal{V}\) converges to a vector \(x\) in \(\mathcal{V}\) . There are two possibilities that suggest themselves: \[\|x_{n}-x\| \to 0 \quad \text{as} \quad n \to \infty; \tag{i}\]\[ (x_{n}-x, y) \to 0 \quad \text{as} \quad n \to \infty, \quad \text{ for each fixed } y \text{ in } \mathcal{V}. \tag{ii}\] If (i) is true, then we have, for every \(y\) , \[|(x_{n}-x, y)| \leq\|x_{n}-x\| \cdot\|y\| \to 0,\] so that (ii) is true. In a finite-dimensional space the converse implication is valid: (ii) \(\implies\) (i). To prove this, let \(\{z_{1}, \ldots, z_{N}\}\) be an orthonormal basis in \(\mathcal{V}\) . (Often in this chapter we shall write \(N\) for the dimension of a finite-dimensional vector space, in order to reserve \(n\) for the dummy variable in limiting processes.) If we assume (ii), then \((x_{n}-x, z_{i}) \to 0\) for each \(i=1, \ldots, N\) . Since ( Section: Completeness , Theorem 2) \[\|x_{n}-x\|^{2}=\sum_{i}|(x_{n}-x, z_{i})|^{2},\] it follows that \(\|x_{n}-x\| \to 0\) , as was to be proved.

Concerning the convergence of vectors (in either of the two equivalent senses) we shall use without proof the following facts. (All these facts are easy consequences of our definitions and of the properties of convergence in the usual domain of complex numbers; we assume that the reader has a modicum of familiarity with these notions.) The expression \(\alpha x+\beta y\) defines a continuous function of all its arguments simultaneously; that is, if \((\alpha_{n})\) and \((\beta_{n})\) are sequences of numbers and \((x_{n})\) and \((y_{n})\) are sequences of vectors, then \(\alpha_{n} \to \alpha\) , \(\beta_{n} \to \beta\) , \(x_{n} \to x\) , and \(y_{n} \to y\) imply that \(\alpha_{n} x_{n}+\beta_{n} y_{n} \to \alpha x+\beta y\) . If \(\{z_{i}\}\) is an orthonormal basis in \(\mathcal{V}\) , and if \(x_{n}=\sum_{i} \alpha_{i n} z_{i}\) and \(x=\sum_{i} \alpha_{i} z_{i}\) , then a necessary and sufficient condition that \(x_{n} \to x\) is that \(\alpha_{i n} \to \alpha_{i}\) (as \(n \to \infty\) ) for each \(i=1, \ldots, N\) . (Thus the notion of convergence here defined coincides with the usual one in \(N\) -dimensional real or complex coordinate space.) Finally, we shall assume as known the fact that a finite-dimensional inner product space with the metric defined by the norm is complete; that is, if \((x_{n})\) is a sequence of vectors for which \(\|x_{n}-x_{m}\| \to 0\) , as \(n, m \to \infty\) , then there is a (unique) vector \(x\) such that \(x_{n} \to x\) as \(n \to \infty\) .