The metric properties of vectors have certain important implications for the metric properties of linear transformations, which we now begin to study.
Definition 1. A linear transformation \(A\) on an inner product space \(\mathcal{V}\) is bounded if there exists a constant \(K\) such that \(\|A x\| \leq K\|x\|\) for every vector \(x\) in \(\mathcal{V}\) . The greatest lower bound of all constants \(K\) with this property is called the norm (or bound ) of \(A\) and is denoted by \(\|A\|\) .
Clearly if \(A\) is bounded, then \(\|A x\| \leq \|A\| \cdot\|x\|\) for all \(x\) . For examples we may consider the cases where \(A\) is a (non-zero) perpendicular projection or an isometry; Section: Perpendicular projections , Theorem 1, and the theorem of Section: Isometries , respectively, imply that in both cases \(\|A\|=1\) . Considerations of the vectors defined by \(x_{n}(t)=t^{n}\) in \(\mathcal{P}\) shows that the differentiation transformation is not bounded.
Because in the sequel we shall have occasion to ccnsider quite a few upper and lower bounds similar to \(\|A\|\) , we introduce a convenient notation. If \(P\) is any possible property of real numbers \(t\) , we shall denote the set of all real numbers \(t\) possessing the property \(P\) by the symbol \(\{t: P\}\) , and we shall denote greatest lower bound and least upper bound by inf (for infimum) and sup (for supremum) respectively. In this notation we have, for example, \[\|A\|=\inf \{K : \|A x\| \leq K\|x\| \text { for all } x\}.\]
The notion of boundedness is closely connected with the notion of continuity. If \(A\) is bounded and if \(\epsilon\) is any positive number, by writing \(\delta = \frac{\epsilon}{\|A\|}\) we make sure that \(\|x - y\| < \delta\) implies that \begin{align} \|Ax - Ay\| &= \|A(x - y)\|\\ &\leq \|A\| \cdot \|x - y\|\\ &< \epsilon; \end{align} in other words boundedness implies (uniform) continuity. (In this proof we tacitly assumed that \(\|A\| \neq 0\) ; the other case is trivial.) In view of this fact the following result is a welcome one.
Theorem 1. Every linear transformation on a finite-dimensional inner product space is bounded.
Proof. Suppose that \(A\) is a linear transformation on \(\mathcal{V}\) ; let \(\{x_{1}, \ldots, x_{N}\}\) be an orthonormal basis in \(\mathcal{V}\) and write \[K_{0}=\max \big\{\|A x_{1}\|, \ldots,\|A x_{N}\|\big\}.\] Since an arbitrary vector \(x\) may be written in the form \(x=\sum_{i}(x, x_{i}) x_{i}\) , we obtain, applying the Schwarz inequality and remembering that \(\|x_{i}\|=1\) , \begin{align} \|A x\| &= \Big\|A\Big(\sum_{i}(x, x_{i}) x_{i}\Big)\Big\| \\ &= \Big\|\sum_{i}(x, x_{i}) A x_{i}\Big\|\\ &\leq \sum_{i}|(x, x_{i})| \cdot\|A x_{i}\| \\ &\leq \sum_{i}\|x\| \cdot\|x_{i}\| \cdot\|A x_{i}\|\\ &\leq K_{0} \sum_{i}\|x\| \\ &= N K_{0}\|x\|. \end{align} In other words, \(K=N K_{0}\) is a bound of \(A\) , and the proof is complete. ◻
It is no accident that the dimension \(N\) of \(\mathcal{V}\) enters into our evaluation; we have already seen that the theorem is not true in infinite-dimensional spaces.
EXERCISES
Exercise 1.
- Prove that the inner product is a continuous function (and therefore so also is the norm); that is, if \(x_{n} \to x\) and \(y_{n} \to y\) , then \((x_{n}, y_{n}) \to(x, y)\) .
- Is every linear functional continuous? How about multilinear forms?
Exercise 2. A linear transformation \(A\) on an inner product space is said to be bounded from below if there exists a (strictly) positive constant \(K\) such that \(\|A x\| \geq K\|x\|\) for every \(x\) . Prove that (on a finite-dimensional space) \(A\) is bounded from below if and only if it is invertible.
Exercise 3. If a linear transformation on an inner product space (not necessarily finite-dimensional) is continuous at one point, then it is bounded (and consequently continuous over the whole space).
Exercise 4. For each positive integer \(n\) construct a projection \(E_{n}\) (not a perpendicular projection) such that \(\|E_{n}\| \geq n\) .
Exercise 5.
- If \(U\) is a partial isometry other than \(0\) , then \(\|U\|=1\) .
- If \(U\) is an isometry, then \(\|U A\|=\|A U\|=\|A\|\) for every linear transformation \(A\) .
Exercise 6. If \(E\) and \(F\) are perpendicular projections, with ranges \(\mathcal{M}\) and \(\mathcal{N}\) respectively, and if \(\|E-F\|<1\) , then \(\operatorname{dim} \mathcal{M}=\operatorname{dim} \mathcal{N}\) .
Exercise 7.
- If \(A\) is normal, then \(\|A^{n}\|=\|A\|^{n}\) for every positive integer \(n\) .
- If \(A\) is a linear transformation on a \(2\) -dimensional unitary space and if \(\|A^{2}\|=\|A\|^{2}\) , then \(A\) is normal.
- Is the conclusion of (b) true for transformations on a \(3\) -dimensional space?