1232 CHAPTER 35. WEAK DERIVATIVES

Now define f (x)≡ limn→∞ x∗n (x). Since each x∗n is linear, it follows f is also linear. Inaddition to this,

| f (x)|= limn→∞|x∗n (x)| ≤ K ||x||

where K is some constant which is larger than all the norms of the x∗n. Such a constantexists because the sequence, {x∗n} was bounded. This proves the lemma.

The lemma implies the following important theorem.

Theorem 35.1.3 Let Ω be a measurable subset of Rn and let { fk} be a bounded sequencein Lp (Ω) where 1 < p≤ ∞. Then there exists a weak ∗ convergent subsequence.

Proof: Since Lp′ (Ω) is separable, this follows from the Riesz representation theorem.Note that from the Riesz representation theorem, it follows that if p < ∞, then the

sequence converges weakly.

35.2 Test Functions And Weak DerivativesIn elementary courses in mathematics, functions are often thought of as things which havea formula associated with them and it is the formula which receives the most attention. Forexample, in beginning calculus courses the derivative of a function is defined as the limitof a difference quotient. You start with one function which tends to be identified with aformula and, by taking a limit, you get another formula for the derivative. A jump in ab-straction occurs as soon as you encounter the derivative of a function of n variables wherethe derivative is defined as a certain linear transformation which is determined not by aformula but by what it does to vectors. When this is understood, it reduces to the usualidea in one dimension. The idea of weak partial derivatives goes further in the direction ofdefining something in terms of what it does rather than by a formula, and extra generalityis obtained when it is used. In particular, it is possible to differentiate almost anything ifthe notion of what is meant by the derivative is sufficiently weak. This has the advantageof allowing the consideration of the weak partial derivative of a function without having toagonize over the important question of existence but it has the disadvantage of not beingable to say much about the derivative. Nevertheless, it is the idea of weak partial deriva-tives which makes it possible to use functional analytic techniques in the study of partialdifferential equations and it is shown in this chapter that the concept of weak derivative isuseful for unifying the discussion of some very important theorems. Certain things whichshold be true are.

Let Ω ⊆ Rn. A distribution on Ω is defined to be a linear functional on C∞c (Ω), called

the space of test functions. The space of all such linear functionals will be denoted byD∗ (Ω). Actually, more is sometimes done here. One imposes a topology on C∞

c (Ω) mak-ing it into a topological vector space, and when this has been done, D ′ (Ω) is defined as thedual space of this topological vector space. To see this, consult the book by Yosida [127]or the book by Rudin [114].

Example: The space L1loc (Ω) may be considered as a subset of D∗ (Ω) as follows.

f (φ)≡∫

Ω

f (x)φ (x)dx