**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