Bounded Linear Transformations
In the context of normed vector spaces we have a topology on our spaces and so it makes sense to ask that maps between them be continuous. In the finite-dimensional case, all linear functions are continuous, so this hasn’t really come up before in our study of linear algebra. But for functional analysis, it becomes much more important.
Now, really we only need to require continuity at one point — the origin, to be specific — because if it’s continuous there then it’ll be continuous everywhere. Indeed, continuity at means that for any
there is a
so that
implies
. In particular, if
, then this means
implies
. Clearly if this holds, then the general version also holds.
But it turns out that there’s another equivalent condition. We say that a linear transformation is “bounded” if there is some
such that
for all
. That is, the factor by which
stretches the length of a vector is bounded. By linearity, we only really need to check this on the unit sphere
, but it’s often just as easy to test it everywhere.
Anyway, I say that a linear transformation is continuous if and only if it’s bounded. Indeed, if is bounded, then we find
so as we let approach
— as
approaches
— the difference between
and
approaches zero as well. And so
is continuous.
Conversely, if is continuous, then it is bounded. Since it’s continuous, we let
and find a
so that
for all vectors
with
. Thus for all nonzero
we find
Thus we can use and conclude that
is bounded.
The least such that works in the condition for
to be bounded is called the “operator norm” of
, which we write as
. It’s straightforward to verify that
, and that
if and only if
is the zero operator. It remains to verify the triangle identity.
Let’s say that we have bounded linear transformations and
with operator norms
and
, respectively. We will show that
works as a bound for
, and thus conclude that
. Indeed, we check that
and our assertion follows. In particular, when our base field is itself a normed linear space (like or
itself) we can conclude that the “continuous dual space”
consisting of bounded linear functionals
is a normed linear space using the operator norm on
.