Expressions for the norm

To facilitate working with the norm of a transformation, we consider the following four expressions: In accordance with our definition of the brace notation, the expression , for example, means the set of all real numbers of the form , considered for all ’s for which .

Since is trivially true with any if , the definition of supremum implies that ; we shall prove that, in fact, . Since the supremum in the expression for is extended over a subset of the corresponding set for (that is, if , then ), we see that ; a similar argument shows that .

For any we consider (so that ); we have . In other words, every number of the set whose supremum is occurs also in the corresponding set for ; it follows that , and consequently that .

Similarly if and , we consider and ; we have and hence, by the argument just used, , so that .

To consolidate our position, we note that so far we have proved that Since it follows that ; we shall complete the proof by showing that . For this purpose we consider any vector for which (so that ); for such an we write and we have In other words, we proved that every number that occurs in the set defining , and is different from zero, occurs also in the set of which is the supremum; this clearly implies the desired result.

The numerical function of a transformation given by satisfies the following four conditions: The proof of the first three of these is immediate from the definition of the norm of a transformation; for the proof of (4) we use the equation , as follows. Since we see that ; replacing by and by , we obtain the reverse inequality.

EXERCISES