The Completeness of ${\cal L}^{2}$

The following statements were discussed in more or less detail in class. If you think you understand a particular statement, then you should attempt to explain it to one another. If you can explain it one sitting in such a way that everyone is convinced of its validity, do not hand in a written version of your explanation. If on the other hand, you cannot explain it one sitting-notes are allowed-then you should put together a convincing argument for the statement and hand in a written version.

1.

(a)

The distance function d² is well-defined on ${\cal L}^{2}(E)$.

(b)

The distance function d² defines a metric on the set ${\cal L}^{2}(E)$.

2.

(a)

If [f], $[g] \in {\cal L}^{2}(E)$, then $[fg] \in {\cal L}^{1}(E)$.

(b)

If $m(E) < \infty$, then $\chi_{E} \in {\cal L}^{2}(E)$ and ${\cal L}^{2}(E) \subset {\cal L}^{1}(E)$.

(c)

If $m(E) = \infty$, then it is not the case that ${\cal L}^{2}(E) \subset {\cal L}^{1}(E)$.

(d)

There exist $f \in {\cal L}^{1}$ that are not in ${\cal L}^{2}(E)$.

3.

We define addition and scalar multiplication of equivalence classes by

[f] + [g] = [f + g]

a[f] = [af]

(a)

These operations are well-defined on ${\cal L}^{2}(E)$.

(b)

With these operations, ${\cal L}^{2}(E)$ is a real vector space.

4.

We defined an inner product on ${\cal L}^{2}(E)$ by $< [f],[g] > = \int_{E}fgdm$, which we used to define the norm of an equivalence class, $\parallel [f] \parallel = \sqrt{<[f],[f]>}$.

(a)

The inner product and norm are well-defined on ${\cal L}^{2}(E)$.

(b)

The inner product satisfies the usual definition of an inner product.

(c)

The Cauchy-Schwarz inequality holds:

$$\mid <[f],[g]> \mid \leq \parallel f \parallel \parallel g \parallel$$

(d)

The triangle inequality holds:

$$\parallel [f] + [g] \parallel \leq \parallel [f] \parallel + \parallel [g] \parallel$$

5.

Now we come to the important result: ${\cal L}^{2}(E)$ is complete.

(a)

i.

If $m(E) < \infty$, then every sequence that is Cauchy in ${\cal L}^{2}(E)$ is also Cauchy in ${\cal L}^{1}(E)$.

ii.

If E is bounded, then ${\cal L}^{2}(E)$ is complete.

(b)

If $m(E) = \infty$ then we wrote E as the infinite union of the sets $E_{k} =E \cap [-k,k]$.

i.

Use a Cantor diagonal process to show that every Cauchy sequence has a convergent subsequence.

ii.

Show that ${\cal L}^{2}(E)$ is complete.

A complete inner product space is called a Hilbert space after the German mathematician David Hilbert (1862-1943). He is acknowledged to be one of the greatest mathematicians of the first half of this century. Part of his legacy to the mathematical community is his list of 23 problems that he posed at the International Congress of Mathematicians in 1900, which have since come to be known as ``the Hilbert problems''. These problems have had a profound impact on mathematics in this century. At the end of the millenium-get used to it, its coming-there will most certainly be an attempt by a committee to produce a list of problems for the coming century.