Some basic definitions

Consider a collection C of elements or “vectors” with the following properties:

1. They can be added. If f and g are in C, so is f + g.

2. f + (g + h) = (f + g) + h, f, g, h ∈ C.

3. There is an element 0 ∈ C such that h + 0 = h for all h ∈ C.

4. For each h ∈ C there is an element −h ∈ C such that h + (− h) = 0.

5. g + h = h + g, g, h ∈ C.

6. For each real number α, αh ∈ C.

7. α(g + h) = αg + αh.

8. (α + β)h = α h + β h.

9. α (βh) = (α β)h.

10. To each h ∈ C there corresponds a real number ∥h∥ with the following properties:

11. ∥αh∥ = |α| ∥h∥.

12. ∥h∥ = 0 if, and only if, h = 0.

13. ∥g + h∥ ≤ ∥g∥ + ∥h∥.

14. If {hn} is a sequence of elements of C such that ∥hn − hm∥ → 0 as m, n → ∞, then there is an element h ∈ C such that ∥hn − h∥ → 0 as n → ∞.

The benefits

If you recall in Chapter 2, we introduced the concept of a pseudogradient. This was done in order to solve differential equations in which the right-hand side was not Lipschitz continuous. These equations came about when we tried to show that if the gradient of a C1 function did not vanish, we could decrease the function. But the equations we wanted to use involved the gradient of the function, which was only known to be continuous and not Lipschitz continuous. Our approach was to substitute another function for the gradient which (a) was Lipschitz continuous and (b) allowed one to decrease the function when the gradient does not vanish. This was done by approximating the gradient by a smoother function. In ℝ2 we used the Heine–Borel theorem to cover bounded sets by a finite number of small balls, construct a Lipschitz continuous function in each ball and then piece them together by means of a partition of unity.

However, in an infinite dimensional Hilbert space this approach does not seem to work. We do not have any difficulty constructing a Lipschitz continuous approximation in a ball or a finite number of balls. But we need to cover the space (or portion of the space) with balls that are locally finite (i.e., any small neighborhood intersects only a finite number of them). This is where the rub is. Can this be done in an arbitrary Hilbert space?

The techniques that can be used to solve nonlinear problems are very different from those that are used to solve linear problems. Most courses in analysis and applied mathematics attack linear problems simply because they are easier to solve. The information that is needed to solve them is not as involved or technical in nature as that which is usually required to solve a corresponding nonlinear problem. This applies not only to the practical material but also to the theoretical background.

As an example, it is usually sufficient in dealing with linear problems in analysis to apply Riemann integration to functions that are piecewise continuous. Rarely is more needed. In considering the convergence of series, uniform convergence usually suffices. In general, concepts from functional analysis are not needed; linear algebra is usually sufficient. A student can go quite far in the study of linear problems without being exposed to Lebesgue integration or functional analysis.

However, there are many nonlinear problems that arise in applied mathematics and sciences that require much more theoretical background in order to attack them. If we couple this with the difficult technical details concerning the corresponding linear problems that are usually needed before one can apply the nonlinear techniques, we find that the student does not come in contact with substantive nonlinear theory until a very advanced stage in his or her studies.