Hilbert Space

5. Hilbert space We have shown that Lp (X, µ) is a Banach space – a complete normed space. I shall next discuss the class of Hilbert spaces, a special class of Banach spaces, of which L2 (X, µ) is a standard example, in which the norm arises from an inner product, just as it does in Euclidean space. An inner product on a vector space V over C (one can do the real case too, not much changes) is a sesquilinear form V ×V �C written (u, v), if u, v ⊂ V . The ‘sesqui-’ part is just linearity in the first variable (5.1) (a1 u1 + a2 u2 , v) = a1 (u1 , v) + a2 (u2 , v), anti-linearly in the second (5.2) (u, a1 v1 + a2 v2 ) = a1 (u, v1 ) + a2 (u, v2 ) and the conjugacy condition (u, v) = (v, u) . (5.3) Notice that (5.2) follows from (5.1) and (5.3). If we assume in addition the positivity condition8 (5.4) (u, u) → 0 , (u, u) = 0 ≤ u = 0 , then ⇒u⇒ = (u, u)1/2 (5.5) is a norm on V , as we shall see. Suppose that u, v ⊂ V have ⇒u⇒ = ⇒v⇒ = 1. Then (u, v) = ei� |(u, v)| for some � ⊂ R. By choice of �, e−i� (u, v) = |(u, v)| is real, so expanding out using linearity for s ⊂ R, 0 ∀ (e−i� u − sv , e−i� u − sv) = ⇒u⇒2 − 2s Re e−i� (u, v) + s2 ⇒v⇒2 = 1 − 2s|(u, v)| + s2 . The minimum of this occurs when s = |(u, v)| and this is negative unless |(u, v)| ∀ 1. Using linearity, and checking the trivial cases u = or v = 0 shows that (5.6) |(u, v)| ∀ ⇒u⇒ ⇒v⇒, � u, v ⊂ V . This is called Schwarz’9 inequality. 8Notice 9No that (u, u) is real by (5.3). ‘t’ in this Schwarz. Using Schwarz’ inequality ⇒u + v⇒2 = ⇒u⇒2 + (u, v) + (v, u) + ⇒v⇒2 ∀ (⇒u⇒ + ⇒v⇒)2 =≤ ⇒u + v⇒ ∀ ⇒u⇒ + ⇒v⇒ � u, v ⊂ V which is the triangle inequality. Definition 5.1. A Hilbert space is a vector space V with an inner product satisfying (5.1) - (5.4) which is complete as a normed space (i.e., is a Banach space). Thus we have already shown L2 (X, µ) to be a Hilbert space for any positive measure µ. The inner product is � f g dµ , (5.7) (f, g) = X since then (5.3) gives ⇒f ⇒2 . Another important identity valid in any inner product spaces is the parallelogram law: (5.8) ⇒u + v⇒2 + ⇒u − v⇒2 = 2⇒u⇒2 + 2⇒v⇒2 . This can be used to prove the basic ‘existence theorem’ in Hilbert space theory. Lemma 5.2. Let C � H, in a Hilbert space, be closed and convex (i.e., su + (1 − s)v ⊂ C if u, v ⊂ C and 0 < s < 1). Then C contains a unique element of smallest norm. Proof. We can certainly choose a sequence un ⊂ C such that ⇒un ⇒ � � = inf {⇒v⇒ ; v ⊂ C} . By the parallelogram law, ⇒un − um ⇒2 = 2⇒un ⇒2 + 2⇒um ⇒2 − ⇒un + um ⇒2 ∀ 2(⇒un ⇒2 + ⇒um ⇒2 ) − 4� 2 where we use the fact that (un + um )/2 ⊂ C so must have norm at least �. Thus {un } is a Cauchy sequence, hence convergent by the assumed completeness of H. Thus lim un = u ⊂ C (since it is assumed closed) and by the triangle inequality |⇒un ⇒ − ⇒u⇒| ∀ ⇒un − u⇒ � 0 So ⇒u⇒ = �. Uniqueness of u follows again from the parallelogram law which shows that if ⇒u� ⇒ = � then ⇒u − u� ⇒ ∀ 2� 2 − 4⇒(u + u� )/2⇒2 ∀ 0 . � The fundamental fact about a Hilbert space is that each element v ⊂ H defines a continuous linear functional by H � u ∈−� (u, v) ⊂ C and conversely every continuous linear functional arises this way. This is also called the Riesz representation theorem. Proposition 5.3. If L : H � C is a continuous linear functional on a Hilbert space then this is a unique element v ⊂ H such that (5.9) Lu = (u, v) � u ⊂ H , Proof. Consider the linear space M = {u ⊂ H ; Lu = 0} the null space of L, a continuous linear functional on H. By the as­ sumed continuity, M is closed. We can suppose that L is not identically / M . Consider zero (since then v = 0 in (5.9)). Thus there exists w ⊂ w + M = {v ⊂ H ; v = w + u , u ⊂ M } . This is a closed convex subset of H. Applying Lemma 5.2 it has a unique smallest element, v ⊂ w + M . Since v minimizes the norm on w + M, ⇒v + su⇒2 = ⇒v⇒2 + 2 Re(su, v) + ⇒s⇒2 ⇒u⇒2 is stationary at s = 0. Thus Re(u, v) = 0 � u ⊂ M , and the same argument with s replaced by is shows that (v, u) = 0 � u ⊂ M . ≥ 0. Consider the element w � = Now v ⊂ w + M , so Lv = Lw = w/Lw ⊂ H. Since Lw � = 1, for any u ⊂ H L(u − (Lu)w � ) = Lu − Lu = 0 . It follows that u − (Lu)w � ⊂ M so if w �� = w � /⇒w � ⇒2 (u, w �� ) = ((Lu)w � , w �� ) = Lu (w � , w � ) = Lu . ⇒w � ⇒2 The uniqueness of v follows from the positivity of the norm. � Corollary 5.4. For any positive measure µ, any continuous linear functional L : L2 (X, µ) � C is of the form � f g dµ , g ⊂ L2 (X, µ) . Lf = X Notice the apparent power of ‘abstract reasoning’ here! Although we seem to have constructed g out of nowhere, its existence follows from the completeness of L2 (X, µ), but it is very convenient to express the argument abstractly for a general Hilbert space.