Real and Complex Analysis (Rudin)

80 REAL AND COMPLEX ANALYSIS
In other words, there is one and only one Xo E E such that IIxoll :S IIxll for
every x E E.
PROOF An easy computation, using only the properties listed in Definition
4.1, establishes the identity
(x and Y E H). (1)
This is known as the parallelogram law: If we interpret Ilxll to be the length
of the vector x, (1) says that the sum of the squares of the diagonals of a
parallelogram is equal to the sum of the squares of its sides, a familiar proposition
in plane geometry.
Let 0 = inf {llxll: x E E}. For any x and Y E E, we apply (1) to tx and ty
and obtain
Since E is convex, (x + Y)/2 E E. Hence
Ilx - yI1 2 :s 211xll 2 + 211yI1 2 - 40 2 (x and Y E E). (3)
If also IIxll = lIylI = 0, then (3) implies x = y, and we have proved the uniqueness
assertion of the theorem.
The definition of 0 shows that there is a sequence {Yn} in E so' that
IIYnll-4O as n-4 00. Replace x and Y in (3) by Yn and Ym. Then, as n-4 00 and
m-4 00, the right side of (3) will tend to 0. This shows that {Yn} is a Cauchy
sequence. Since H is complete, there exists an Xo E H so that Yn-4 Xo, i.e.,
llYn - xoll-4 0, as n-4oo. Since Yn E E and E is closed, Xo E E. Since the
norm is a continuous function on H (Theorem 4.6), it follows that
IIxoll = lim llYn II = 0.
n-+ co
4.11 Theorem Let M be a closed subspace of a Hilbert space H.
(a) Every x E H has then a unique decomposition
x = Px + Qx
into a sum of Px E M and Qx E Ml..
(b) Px and Qx are the nearest points to x in M and in Ml., respectively.
(c) The mappings P: H -4 M and Q: H -4 Ml. are linear.
(d) IIxl12 = IIPxl1 2 + IIQxI12.
Corollary If M ¥- H, then there exists Y E H, Y ¥- 0, such that Y .1 M.
P and Q are called the orthogonal projections of H onto M and Ml..
(2)
IIII