1. Week 2 Introduction to Hilbert spaces Review of vector spaces
Review of inner product spaces
Metric spaces
Complete metric spaces, Hilbert spaces

2. 1. Review of vector spaces and inner product spaces
۞ Let V be a set on which operations of vector addition (VA) and vector-by-scalar multiplication (VSM) are defined, i.e.:
with each elements x and y in V, one can associate a unique element x + y, and
with each element x in V and a scalar α, one can associate a unique element αx.
The set V, together with the operations, is said to form a vector space if and only if the following conditions (axioms) are satisfied:

4. Example 1: The importance of the closure axioms (C1 & C2)
Consider the set of ordered pairs of numbers such that
The usual (Euclidean) vector addition for V doesn’t satisfy C1, e.g.
but

5. Example 2:
(a) The set of all complex numbers, with the usual addition and multiplication by a real number, form a vector space (they satisfy all axioms).
(b) The set of polynomials of degree less than 3, with the usual operations of their addition and multiplication by a number, form a vector space (denoted usually by P3).
(c) The set of all polynomials with... form a vector space (denoted usually by P∞).
(d) The set C(a, b) of functions continuous in an interval (a, b).

6. For some vector spaces, there’s a one-to-one correspondence with Rn, e
For some vector spaces, there’s a one-to-one correspondence with Rn, e.g.

7. Example 3:
Let V be the set of all ordered pairs of real numbers VA and VSM defined by
Is V a vector space with these operations?
Soln:
No, because the above VSM violates several axioms, e.g. A6.

8. Example 4:
Let V be the set of all ordered pairs of real numbers VA and VSM defined by
Is V a vector space with these operations?
Soln:
No, because the above VA violates several axioms, e.g. A3.
Indeed, consider a pair [x1, x2] with x2 ≠ 0 and add to it the zero vector 0 = [0, 0]...

9. Note that with the vector 0 doesn’t have to be [0, 0]
Note that with the vector 0 doesn’t have to be [0, 0]. In principle, it can be any pair [a, b]. But no matter what a and b one chooses, A3 doesn’t hold (because the VA always changes x2 into zero).