Real and Complex Analysis (Rudin)

ELEMENTARY HILBERT SPACE THEORY 81
PROOF As regards the uniqueness in (a), suppose that x' + y' = x" + y" for
some vectors x', x" in M and y', y" in Ml.. Then
x' - x" = y" - y'.
Since x' - x" E M, y" - y' E Ml., and M n Ml. = {O} [an immediate conse­
"quence of the fact that (x, x) = 0 implies x = 0], we have x" = x', y" = y'.
To prove the existence of the decomposition, note that the set
x + M = {x + y: y E M}
is closed and convex. Define Qx to be the element of smallest norm in
x + M; this exists, by Theorem 4.10. Define Px = x - Qx.
Since Qx E x + AI, it is clear that Px E M. Thus P maps H into M.
To prove that Q maps H into Ml. we show that (Qx, y) = 0 for all y E M.
Assume Ilyll = 1, without loss of generality, and put z = Qx. The minimizing
property of Qx shows that
(z, z) = IIzll2 ~ liz - lXyll2 = (z - IXY, z - IXY)
for every scalar IX. This simplifies to
o ~ -1X(y, z) - ,x(z, y) + 1X,x.
With IX = (z, y), this gives 0 ~ - I (z, y) 1 2 , so that (z, y) = O. Thus Qx E Ml..
We have already seen that Px E M. If y E M, it follows that
Ilx - yl12 = IIQx + (Px - y)11 2 = IIQxl12 + IIPx _ yl12
which is obviously minimized when y = Px.
We have now proved (a) and (b). If we apply (a) to x, to y, and to
IXX + [Jy, we obtain
P(IXX + [Jy) -
IXPX - [JPy = IXQX + [JQy - Q(IXX + [Jy).
The left side is in M, the right side in Ml.. Hence both are 0, so P and Q are
linear.
Since Px .l Qx, (d) follows from (a).
To prove the corollary, take x E H, x ¢ M, and put y = Qx. Since
Px E M, x #: Px, hence y = x - Px #: O.
IIII
We have already observed that x-+ (x, y) is, for each y E H, a continuous
linear functional on H. It is a very important fact that all continuous linear
functionals on H are of this type.
4.12 Theorem If L is a continuous linear functional on H, then there is a
unique y E H such that
Lx = (x, y) (x E H). (1)