Hilbert Space Methods in Partial Differential Equations

Chapter I

Elements of Hilbert Space

1Linear Algebra

We begin with some notation. A function F with domain dom(F) = A and range Rg(F) a subset of B is denoted by F : A → B. That a point x ∈ A is mapped by F to a point F (x) ∈ B is indicated by x F(x). If S is a subset of A then the image of S by F is F(S) = {F(x) : x ∈ S}. Thus Rg(F) = F(A). The pre-image or inverse image of a set T ⊂ B is F−1(T) = {x ∈ A : F(x) ∈ T}. A function is called injective if it is one-to-one, surjective if it is onto, and bijective if it is both injective and surjective. Then it is called, respectively, an injection, surjection, or bijection.

will denote the field of scalars for our vector spaces and is always one of (real number system) or (complex numbers). The choice in most situations will be clear from the context or immaterial, so we usually avoid mention of it.

The strong inclusion K ⊂⊂ G between subsets of Euclidean space n means K is compact, G is open, and K ⊂ G. If A and B are sets, their Cartesian product is given by A × B = {[a, b] : a ∈ A, b ∈ B}. If A and B are subsets of (or any other vector space) their set sum is A + B = {a + b : a ∈ A, b ∈ B}.

1.1

A linear space over the field is a non-empty set V of vectors with a binary operation addition + : V × V → V and a scalar multiplication : × V → V such that (V, +) is an Abelian group, i.e.,

and we have

We shall suppress the symbol for scalar multiplication since there is no need for it.

Examples. (a) The set of n-tuples of scalars is a linear space over . Addition and scalar multiplication are defined coordinatewise:

(b) The set of functions f : X → is a linear space, where X is a non-empty set, and we define (f1 + f2)(x) = f1(x) + f2(x), (αf)(x) = αf (x), x ∈ X.

(c) Let G ⊂ n be open. The above pointwise definitions of linear operations give a linear space structure on the set C(G, ) of continuous f : G → . We normally shorten this to C(G).

(d) For each n-tuple α = (α1, α2, …, αn) of non-negative integers, we denote by Da the partial derivative

of order |α| = α1 + α 2 + … + αn. The sets Cm(G) = {f ∈ C(G) : Dα f ∈ C(G) for all α, |α| ≤ m}, m ≥ 0, and C∞G = ∩m≥1 Cm(G) are linear spaces with the operations defined above. We let Dθ be the identity, where θ = (0,0,… ,0), so C⁰(G) = C(G).

(e) For f ∈ C(G), the support of f is the closure in G of the set {x ∈ G : f(x) ≠ 0} and we denote it by supp(f). C0(G) is the subset of those functions in C(G) with compact support. Similarly, we define

and .

(f) If f : A → B and C ⊂ A, we denote f|C the restriction of F to C. We obtain useful linear spaces of functions on the closure as follows:

These spaces play a central role in our work below.

1.2

A subset M of the linear space V is a subspace of V if it is closed under the linear operations. That is, x + y ∈ M whenever x,y ∈ M and ax ∈ M for each a ∈ and x ∈ M. We denote that M is a subspace of V by M ≤ V. It follows that M is then (and only then) a linear space with addition and scalar multiplication inherited from V.

Examples. We have three chains of subspaces given by

Moreover, for each k as above, we can identify with that Φ ∈ Ck ( ) obtained by defining Φ to be equal to φ on G and zero on ∂G, the boundary of G. Likewise we can identify each Φ ∈ Ck( ) with Φ|G ∈ CK(G). These identifications are compatible and we have Ck(G).

1.3

We let M be a subspace of V and construct a corresponding quotient space. For each x ∈ V, define a coset = {y ∈ V : y − x ∈ M} = {x + m : m ∈ M}. The set V/M = { : x ∈ V} is the quotient set. Any y ∈ is a representative of the coset and we clearly have y ∈ if and only if x ∈ ŷ if and only if = ŷ. We shall define addition of cosets by adding a corresponding pair of representatives and similarly define scalar multiplication. It is necessary to first verify that this definition is unambiguous.

Lemma If x1, x2 ∈ , y1, y2 ∈ ŷ and α ∈ , then and

The proof follows easily, since M is closed under addition and scalar multiplication, and we can define and . These operations make V/M a linear space.

Examples. (a) Let V = ² and M = {(0, x2) : x2 ∈ }. Then V/M is the set of parallel translates of the x2-axis, M, and addition of two cosets is easily obtained by adding their (unique) representatives on the x1-axis.

(b) Take V = C(G). Let x0 ∈ G and M = {φ ∈ C(G) : φ(x0) = 0}. Write each φ ∈ V in the form φ(x) = (φ(x) − φ(x0)) + φ(x0). This representation can be used to show that V/M is essentially equivalent (isomorphic) to .

(c) Let V = C( ) and M = C0(G). We can describe V/M as a space of boundary values. To do this, begin by noting that for each K ⊂⊂ G there is a ∈ C0(G) with = 1 on K. (Cf. Section II.1.1.) Then write a given φ ∈ C ( ) in the form

where the first term belongs to M and the second equals φ in a neighborhood of ∂G.

1.4

Let V and W be linear spaces over . A function T : V → W is linear if

That is, linear functions are those which preserve the linear operations. An isomorphism is a linear bijection. The set {x ∈ V : Tx = 0} is called the kernel of the (not necessarily linear) function T :V → W and we denote it by K(T).

Lemma If T : V → W is linear, then K(T) is a subspace of V, Rg(T) is a subspace of W, and K(T) = {θ} if and only if T is an injection.

Examples. (a) Let M be a subspace of V. The identity im : M→ V is a linear injection x x and its range is M.

(b) The quotient map qM : V → V/M, x , is a linear surjection with kernel K (qM) = M.

(c) Let G be the open interval (a, b) in and consider D ≡ d/dx: V → C( ), where V is a subspace of C¹( ). If V = C¹( ), then D is a linear surjection with K(D) consisting of constant functions on . If V = {φ ∈ C¹( ) : φ(α) = 0}, then D is an isomorphism. Finally, if V = {φ ∈ C¹( ) : φ(α) = φ(b) = 0}, then

Our next result shows how each linear function can be factored into the product of a linear injection and an appropriate quotient map.

Theorem 1.1 Let T : V → W be linear and M be a subspace of K(T). Then there is exactly one function : V/M → W for which qM = T, and is linear with Rg( ) = Rg(T). Finally, is injective if and only if M = K(T).

Proof: If x1, x2 ∈ , then x1 − x2 ∈ M ⊂ K(T), so T(x1) = T(x2). Thus we can define a function as desired by the formula ( ) = T(x). The uniqueness and linearity of follow since qM is surjective and linear. The equality of the ranges follows, since qM is surjective, and the last statement follows from the observation that K(T) ⊂ M if and only if v ∈ V and ( ) = 0 imply

An immediate corollary is that each linear function T : V → W can be factored into a product of a surjection, an isomorphism, and an injection:

A function T : V → W is called conjugate linear if

Results similar to those above hold for such functions.

1.5

Let V and W be linear spaces over and consider the set L(V, W) of linear functions from V to W. The set WV of all functions from V to W is a linear space under the pointwise definitions of addition and scalar multiplication (cf. Example 1.1(b)), and L(V, W) is a subspace.

We define V to be the linear space of all conjugate linear functionals from V → . V is called the algebraic dual of V. Note that there is a bijection f of (V, ) onto V , where is the functional defined by for x ∈ V and is called the conjugate of the functional f : V → . Such spaces provide a useful means of constructing large linear spaces containing a given class of functions. We illustrate this technique in a simple situation.

Example. Let G be open in n and x0 ∈ G. We shall imbed the space C(G) in the algebraic dual of C0(G). For each f ∈ C(G), define Tf ∈ C0(G) by

Since , the Riemann integral is adequate here. An easy exercise shows that the function f Tf : C(G) → C0(G) is a linear injection, so we may thus identify C(G) with a subspace of C0(G) . This linear injection is not surjective; we can exhibit functionals on C0(G) which are not identified with functions in C(G). In particular, the Dirac functional ∂x0 defined by

cannot be obtained as Tf for any f ∈ C(G). That is, Tf = δx0 implies that f(x) = 0 for all x ∈ G, x ≠ x0, and thus f = 0, a contradiction.

2Convergence and Continuity

The absolute value function on R and modulus function on C are denoted by | · |, and each gives a notion of length or distance in the corresponding space and permits the discussion of convergence of sequences in that space or continuity of functions on that space. We shall extend these concepts to a general linear space.

2.1

A seminorm on the linear space V is a function p : V → for which p(αx) =|α|p(x) and p(x + y) ≤ p(x) + p(y) for all α ∈ and x,y ∈ V. The pair V,p is called a seminormed space.

Lemma 2.1 If V, p is a seminormed space, then

(a) |p(x) − p(y)| ≤ p(x − y), x,y ∈ V,

(b) p(x) ≥ 0 , x ∈ V , and

(c) the kernel K(p) is a subspace of V.

(d) If T ∈ L(W, V), then p T : W → is a seminorm on W.

(e) If pj is a seminorm on V and αj ≥ 0, 1 ≤ j ≤ n, then is a seminorm on V.

Proof: We have p(x) = p(x−y+y) ≤ p(x−y)+p(y) so p(x) − p(y) ≤ p(x−y).Similarly, p(y) − p(x) ≤ p(y − x) = p(x − y), so the result follows. Setting y = 0 in (a) and noting p(0) = 0, we obtain (b). The result (c) follows directly from the definitions, and (d) and (e) are straightforward exercises.

If p is a seminorm with the property that p(x) > 0 for each x ≠ θ, we call it a norm.

Examples. (a) For 1 ≤ k ≤ n we define seminorms on by pk(x) =

and rk(x) = max{|xj| : 1 ≤ j ≤ k}. Each of pn, qn and rn is a norm.

(b) If J ⊂ X and f ∈ , we define pJ(f) = sup{|f(x)| : x ∈ J}. Then for each finite J ⊂ X, pJ is a seminorm on .

(c) For each K ⊂⊂ G, pK is a seminorm on C(G). Also, is a norm on C( ).

(d) For each j, 0 ≤ j ≤ k, and K ⊂⊂ G we can define a seminorm on Ck(G) by pj,K(f) = sup{|Dα f(x)| : x ∈ K, |α| ≤ j}. Each such pj,G is a norm on Ck( ).

2.2

Seminorms permit a discussion of convergence. We say the sequence {xn} in V converges to x ∈ V if limn→∞p(xn − x) = 0; that is, if {p(xn − x)} is a sequence in converging to 0. Formally, this means that for every ε > 0 there is an integer N ≥ 0 such that p(xn − x) < ε for all n ≥ N. We denote this by xn → x in V,p and suppress the mention of p when it is clear what is meant.

Let S ⊂ V. The closure of S in V,p is the set = {x ∈ V : xn → x in V,p for some sequence {xn} in S}, and S is called closed if S = . The closure of S is the smallest closed set containing, and if S ⊂ K = then ⊂ K.

Lemma Let V,p be a seminormed space and M be a subspace of V. Then is a subspace of V.

Proof: Let x,y ∈ . Then there are sequences xn,yn ∈ M such that xn → x and yn → y in V,p. But p((x+y) − (xn+yn)) ≤ p(x−xn)+p(y−yn) → 0 which shows that (xn + yn) → x + y. Since xn + yn ∈ M, all n, this implies that x+y ∈ . Similarly, for α ∈ we have p(αx−αxn) = |α|p(x − xn) → 0, so αx ∈ .

2.3

Let V,p and W,q be seminormed spaces and T : V → W (not necessarily linear). Then T is called continuous at x ∈ V if for every ε > 0 there is a δ > 0 for which y ∈ V and p(x − y) < δ implies q(T(x) − T(y)) < ε. T is continuous if it is continuous at every x ∈ V.

Theorem 2.2 T is continuous at x if and only if xn → x in V,p implies Txn → Tx in W, q.

Proof: Let T be continuous at x and ε > 0. Choose δ > 0 as in the definition above and then N such that n ≥ N implies p(xn − x) < δ, where xn → x in V,p is given. Then n ≥ N implies q(Txn − Tx) < ε, so Txn → Tx in W, q.

Conversely, if T is not continuous at x, then there is an ε > 0 such that for every n ≥ 1 there is an xn ∈ V with p(xn − x) < 1/n and q(Txn − Tx) ≥ ε. That is, xn → x in V,p but {Txn} does not converge to Tx in W, q.

We record the facts that our algebraic operations and seminorm are always continuous.

Lemma If V,p is a seminormed space, the functions (α, x) αx : × V → V, (x, y) x + y : V × V V, and p : V are all continuous.

Proof: The estimate p(αx − αnxn) ≤ |α − αn|p(x) + |αn|p(x − xn) implies the continuity of scalar multiplication. Continuity of addition follows from an estimate in the preceding Lemma, and continuity of p follows from the Lemma of 2.1.

Suppose p and q are seminorms on the linear space V. We say p is stronger than q (or q is weaker