For most of our applications, we will use one of three possible vector norms as already identified. The question that faces us is what are the compatible operator norms induced by these vector norms.We will answer the question once in detail and leave the other two for discussion later.
Proof.
Showing this requires two steps. First we must show that the supremum in the definition of the operator norm is bounded above and does exist. That is,
(Remember that it could easily be unbounded). Second we must show the supremum has the desired value
To this end, let be defined as above. For
, we have
using the triangle inequality for absolute value. We can interchange the order
of summation and use our definition of
to obtain
Hence
or
Since
was arbitrary,
Which shows the supremum is bounded above, which was our first part.
Now that we have this bound, what remains to be shown is that the is a vector
such that
This will imply that
is the maximum. (Remember that this bound will only allow
to be strictly greater than the supremum or it is equal to it. If we show it is equal to it, then we have eliminated the strictly greater than case. Also note that there may be more than vector that achieve this maximum value. That doesn't matter. Only the value of the maximum matters.)
To find such a vector
, we note that there is a number such that
Let
be the Euclidean basis vector. Let
. Then
And we are done.
So the operator norm induced by the 1-norm is the maximum value of the sum of the absolute value of the entries in a column.
Let use examine the other induced operator norms.
On the basis of these examples, you might guess that the operator norm induced by the standard Euclidean norm is