Metric and Hilbert Spaces
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updated: 4 November 2014
Lecture 29: Norms of linear operators
Let and be vector spaces over
A linear operator is a function such that
(a) |
if then
|
(b) |
if and then
|
Let and be normed vector spaces over where
is or
A bounded linear operator from to is a linear operator
such that
there exists such that
if then
The operator norm of a bounded linear operator is
HW: Let and be normed vector spaces. Show that
with norm
given by is a normed vector space.
Let and be normed vector spaces. If is a Banach space then
is a Banach space.
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Assume is a Banach space.
To show: is complete.
To show: If
is a Cauchy sequence in then
converges to
Assume is a
Cauchy sequence in
To show: There exists such that
Let be given by
|
To show: |
(a) |
exists in
|
(b) |
|
(c) |
|
|
From here it is possible to follow the proof in the Rubinstein notes.
|
|
Let and be normed vector spaces and let
be a linear operator. Then
(a) |
if and only if
is continuous.
|
(b) |
is continuous if and only if is uniformly continuous.
|
![](plus.gif) |
![](minus.gif) |
Proof. |
|
To show: |
(a) |
If then
is uniformly continuous.
|
(b) |
If is uniformly continuous then is continuous.
|
(c) |
If is continuous then
|
|
(b) |
was a direct consequence of the way that we set up the definitions of 'uniformly continuous' and 'continuous'.
|
(a) |
Assume
To show: is uniformly continuous.
To show: If then there exists
such that if
and
then
This is a consequence of the computation
|
(c) |
To show: If is continuous then
Assume is continuous.
To show: is bounded.
To show: There exists such that if
then
Since is continuous, is continuous at
So, there exists such that if
and
then
Let
To show: If then
Assume
Let
so that
Then
So
So is bounded.
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Notes and References
These are a typed copy of Lecture 29 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on September 16, 2014.
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