1. Week 3 2. Review of inner product spaces
۞ An inner product on a vector space V is an operation that assigns to each pair of vectors x and y in V a real number <x, y>, and satisfies the following conditions (axioms):
۞ A vector space with an inner product is called an inner product space.

2. Example 1:
The standard inner product (IP) for ℝn is the scalar (dot) product:
We can also define a different IP, e.g.
where w1, w2, and w3 are positive real numbers (weights).
Q: Why do weights have to be positive? Which axiom would be violated if, say, one of the weights is negative or zero?

3. Example 2:
The usual inner product for the space C[a, b] of real continuous functions defined on a closed interval [a, b] is
(1)
۞ Vectors x and y of an inner product space are called orthogonal if <x, y> = 0.
Example 3:
Show that the vectors f = [sin x] and g = [cos x] are orthogonal in C[–π, π] with respect to inner product (1).

4. ۞ The norm of a vector x in an inner product space is a real positive number denoted by || x || and equal to
Theorem 1: The Cauchy–Schwartz Lemma (CSL)
In an inner space, any vectors x and y satisfy
where the equality occurs if and only if x and y are parallel, i.e. if either y = const × x or x = const × y.

5. Proof:
If y = 0, the CSL holds. Next, consider the case y ≠ 0 and introduce
Consider the (obviously correct) inequality
(2)
hence,
Expand the above expression using Axiom (iv) to obtain

6. ...hence, recalling the definition of λ,
(3)
which is equivalent to the Cauchy–Schwartz inequality.
It remains to prove that...

7. (*) If x and y are parallel, the CSL holds with “=” instead of “≥”.
(**) If the CSL holds with “=”, then x and y are parallel.
To prove (*), let x = αy in the CSL and see what happens.
To prove (**) observe that replacing “≥” with “=” in CLS implies the same change in (3) – and, hence, in all the previous inequalities including (2) – thus,
hence,
hence, x is parallel to y, QED (quod erat demonstrandum).

8. ۞ The angle θ between vectors x and y is defined by
The existence of a real θ is guaranteed by CSL (how?).
Thus, using inner product, one can define the angle between any objects which can be interpreted as vectors, e.g. functions.

9. 3. Metric spaces
“Metric” is a fancy word for “distance”. A metric space is a set where, for each pair of elements x and y, we have defined the distance d(x, y) between them.
۞ A metric on a set M is a function d: M×M → ℝ such that, for any x, y, and z in M,

10. Comment:
Axiom (A) is superfluous, as it follows from the other three axioms.
Indeed, consider (D) with z = x,
then take into account (B):
then take into account (C),
then cancel 2 and thus obtain (A).

11. Comment:
Metric spaces are not directly related to vector spaces.
A vector space is a metric space only if we define d(x, y) for it.
A metric space is a vector space only if we define a vector addition and a vector-by-scalar multiplication for it.
Still, we’ll consider many examples which are both vector and metric spaces.
Example 4:
ℝ2 with
make a metric space.

12. Comment:
Metric spaces and inner product spaces are related (unlike the former and vector spaces).
Theorem 2:
An inner product space is always a metric space with
Proof: see Q4.3.
Comment:
A metric space is not necessarily an inner product space (for one thing, the former doesn’t have to be a vector space, whereas the latter does).

13. Example 5:
The set C[a, b] of all continuous functions defined on a closed interval [a, b] and
make a metric space.
Comments:
Q1: How can we be sure that the integral in d(f, g) exists?
Q2: Would it exist if we considered C(a, b) instead of C[a, b]?
A2: In C(a, b), functions are allowed to tend to infinity as x → a or x → b, and for such functions the metric can be infinite.

14. Proof (of the statement in Example 5):
Axioms (A), (C), and the “ part” of Axiom (B) obviously hold.
To prove the “ part” of Axiom (B), let
and observe that f(x) and g(x) can differ only at isolated points of x – which they however can’t, as they are both continuous.

15. To prove that Axioms (D) holds, consider the (obviously correct) inequality:
(4)
which holds when p and q are either real numbers or real functions.
Assuming the latter, let
where f(x), g(x), and h(x) are certain functions.
Then, (4) becomes
(5)

16. Integrating (5) with respect to x over [a, b] and recalling this example’s definition of metric, we obtain
which is Axiom (D) (with x, y, and z replaced with f, g, and h).

17. (a) {xn = (n + 1)/n}, (b) {xn = 1/n2}.
Quick review of limits of sequences in ℝ
۞ A sequence or real numbers {xn} (n = 1, 2, 3...) is said to converge to a limit L if
Comment:
Here, N depends on (is a function of) ε.
Example 6:
Prove that the sequence {xn = 1/n} converges to x = 0.
Example 7:
Find the limits of the following sequences:
(a) {xn = (n + 1)/n}, (b) {xn = 1/n2}.

18. ۞ A sequence {xn} in a metric space with a metric d(x, y) is said to converge to a limit L if