Basic Analysis – Gently Done Topological Vector Spaces

124 Basic Analysis
Remark 10.27 It is sometimes easier to check that a map T has a closed graph
than to check that it is continuous. For the latter, it is necessary to show that
if x n → x, then Tx n does, in fact, converge and has limit equal to Tx. To show
that T has a closed graph, one starts with the hypotheses that both x n and Tx n
converge, the first to x and the second to some y. All that remains is to show that
Tx = y.
The following application to operators on a Hilbert space is of interest. It says
that a symmetric operator defined on the whole Hilbert space is bounded. This
is of interest in quantum mechanics where symmetric operators (or self-adjoint
operators, to be precise) are used to represent physically observable quantities. It
turns out that these are often unbounded operators. The following theorem says
that it is no use trying to define such objects on the whole space. Instead one uses
dense linear subspaces as domains of definition for unbounded operators.
Theorem 10.28 (Hellinger-Toeplitz) Suppose that T : H → H is a linear operator
on a Hilbert space H such that
〈Tx,y〉 = 〈x,Ty〉
for every x,y ∈ H. Then T is continuous.
Proof A Hilbert space is a Banach space, so is complete. We need only show that
T has closed graph. So suppose that (x n ,Tx n ) → (x,y) in H×H. For any z ∈ H,
However, Tx n → y and so
〈Tx n ,z〉 = 〈x n ,Tz〉 → 〈x,Tz〉 = 〈Tx,z〉.
〈y,z〉 = 〈Tx,z〉
and we deduce that y = Tx. It follows that Γ(T) is closed and so T is a continuous
linear operator on H.
We shall end this chapter with a brief discussion of projections. Let X be a
linear space and suppose that V and W are subspaces of X such that V ∩W = {0}
and X = span{V,W}. Then any x ∈ X can be written uniquely as x = v + w
with v ∈ V and w ∈ W. In other words, X = V ⊕W. Define a map P : X → V
by Px = v, where x = v +w ∈ X, with v ∈ V and w ∈ W, as above. Evidently,
P is a well-defined linear operator satisfying P 2 = P. P is called the projection
onto V along W. We see that ranP = V (since Pv = v for all v ∈ V), and also
kerP = W (since if x = v + w and Px = 0 then we have 0 = Px = v and so
x = w ∈ W).
Conversely, suppose that P : X → X is a linear operator such that P 2 = P,
that is, P is an idempotent. Set V = ranP and W = kerP. Evidently, W is a
Department of Mathematics King’s College, London