Reference: Introductory Real Analysis by Kolmogorov and Fomin.
Reference: Analysis by Walter Rudin.
Reference: Introduction to Analysis by Rosenlicht.
Reference: Topology by Hocking and Young.
The problems proposed were as follows:
- Given a metric space
, prove that (1a)
where
; (1b)
where
.
- Verify that
. Deduce the Cauchy-Schwarz inequality from the above identity. (refer second last blog from here). For convenience, we reproduce it here again: consider the set of all ordered n-tuples such that
and
and
and let
,
where
: then Cauchy Schwarz inequality tells us that :
- Verify that
.
Deduce Schwarz’s inequality from this identity. (refer second last blog from here). To aid the memory, we present Schwarz’s inequality below:
4. Identify the fallacy in example 10, if ? (refer second last blog from here). Hint: Show that Minkowski’s inequality fails for
. To help recall, we are reproducing Minkowski’s inequality below: Consider again the set of all ordered n-tuples, and let
, and
and
and again further let
and
, where
then Minkowski’s inequality tells us that:
5. Prove that the following metric : Consider two points and
with the metric function:
is the limiting case of the following metric: consider the set of all ordered n-tuples of real numbers with the metric function now defined by:
in the sense that
6. Starting from the following inequality: , deduce Holder’s integral inequality as given by:
where
valid for any function
and
such that the integrals on the right exist.
7. Use Holder’s integral inequality to prove Minkowski’s integral inequality:
where .
8. Exhibit an isometry between the spaces and
.
9. Verify that the following is a metric space: all bounded infinite sequences of elements of
, with
10. Verify that is a metric space where
when
and is
when
. Illustrate by diagrams in the plane
or
what the open balls of this metric space are.
11. Show that a one-to-one transformation of a space S onto a space T is a homeomorphism if and only if both f and
are continuous.
The solutions are as follows:
Problem 1:
- Given a metric space
, prove that (1a)
where
; (1b)
where
.
Solution 1:
Part I: TPT:
Proof: We know that: , which also means that,
, that is,
. On the other hand, we also know that
, which also means that,
, that is,
. Combining the two smallish results, we get
. QED.
Part II: TPT: . Proof : We know that
, that is,
. On the other hand, we know that
, that is,
, that is,
. Combining the two smallish results, we get
where
. QED.
PS: Now that the detailed proof is presented, try to interpret the above two results “geometrically.”
Problem 2 was:
Verify that . Deduce the Cauchy-Schwarz inequality from the above identity. (refer second last blog from here). For convenience, we reproduce it here again: consider the set of all ordered n-tuples such that
and
and
and let
,
where
: then Cauchy Schwarz inequality tells us that :
.
Proof of problem 2: We present a proof based on the principle of mathematical induction. Let us check the proposition for : Then,
. So now let the proposition be true for
. Then, the following holds true:
.
Now, we want to prove the proposition for , that is, we need to prove that:
.
Now, let us see what we need really; we need to add some(same) terms to both sides of the proposition for and manipulate to derive the proposition for
:
Further notice that
Comparing the algebraic statements for proposition for and
, we can clearly see which terms are to be added to both sides of the proposition for
,
(please fill in the few missing last details…a good deal of algebraic manipulation…some concentrated effort …:-))
From this the Cauchy Schwarz inequality follows naturally.
PS: I have no idea how to solve question 3. If any reader helps, I would be obliged.
PS: I have yet to try the other questions.
Cheers,
Nalin Pithwa