Norm

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 on an inner product space is bounded if there exists a constant such that for every vector in . The greatest lower bound of all constants with this property is called the norm (or bound ) of and is denoted by .

Clearly if is bounded, then for all . For examples we may consider the cases where 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 . Considerations of the vectors defined by in 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 , we introduce a convenient notation. If is any possible property of real numbers , we shall denote the set of all real numbers possessing the property by the symbol , 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,

The notion of boundedness is closely connected with the notion of continuity. If is bounded and if is any positive number, by writing we make sure that implies that in other words boundedness implies (uniform) continuity. (In this proof we tacitly assumed that ; 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 is a linear transformation on ; let be an orthonormal basis in and write Since an arbitrary vector may be written in the form , we obtain, applying the Schwarz inequality and remembering that , In other words, is a bound of , and the proof is complete. ◻

It is no accident that the dimension of 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 and , then .
Is every linear functional continuous? How about multilinear forms?

Exercise 2. A linear transformation on an inner product space is said to be bounded from below if there exists a (strictly) positive constant such that for every . Prove that (on a finite-dimensional space) 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 construct a projection (not a perpendicular projection) such that .

Exercise 5.

If is a partial isometry other than , then .
If is an isometry, then for every linear transformation .

Exercise 6. If and are perpendicular projections, with ranges and respectively, and if , then .

Exercise 7.

If is normal, then for every positive integer .
If is a linear transformation on a -dimensional unitary space and if , then is normal.
Is the conclusion of (b) true for transformations on a -dimensional space?

EXERCISES

Quick Links

Social Media