Real and Complex Analysis (Rudin)

EXAMPLES OF BANACH SPACE TECHNIQUES 99
If one of these sets, say V N , fails to be dense in X, then there exist an
Xo E X and an r > 0 such that Ilxll s; r implies Xo + x ¢ V N ; this means that
q>(xo + x) s; N, or
IIA",(xo + x)11 s; N (5)
for alIa E A and all x with Ilxll s; r. Since x = (xo + x) - xo, we then have
IIA",xll s; IIA",(xo + x)11 + IIA",xoll s; 2N, (6)
and it follows that (1) holds with M = 2N/r.
The other possibility is that every V. is dense in X. In that case, n v. is a
dense G 6 in X, by Baire's theorem; and since q>(x) = 00 for every x E n v.,
the proof is complete. / / / /
5.9 The Open Mapping Theorem Let U and V be the open unit balls of the
Banach spaces X and Y. To every bounded linear transformation A of X onto
Y there corresponds a () > 0 so that
A(U) => {)V. (1)
Note the word" onto" in the hypothesis. The symbol () V stands for the set
y: y E V}, i.e., the set of all y E Y with Ilyll < ().
It follows from (1) and the linearity of A that the image of every open ball in
. with center at xo, say, contains an open ball in Y with center at Axo. Hence
e image of every open set is open. This explains the name of the theorem.
Here is another way of stating (1): To every y with Ilyll < () there corresponds
x with Ilxll < 1 so that Ax = y.
PROOF Given y E Y, there exists an x E X such that Ax = y; if Ilxll < k, it
follows that y E A(kU). Hence Y is the union of the sets A(kU), for
k = 1, 2, 3, " " Since Y is complete, the Baire theorem implies that there is a
nonempty open set W in the closure of some A(kU). This means that every
point of W is the limit of a sequence {AXi}, where Xi E kU; from now on, k
and Ware fixed.
Choose Yo E W, and choose '1 > 0 so that Yo + YEW if Ilyll < '1. For
any such y there are sequences {x;}, {xn in kU such that
Axi- Yo + y (i- 00). (2)
Setting Xi = xi - x;, we have Ilxili < 2k and AXi- y. Since this holds for
every y with Ilyll < '1, the linearity of A shows that the following is true, if
() = '1/2k:
To each y E Y and to each E > 0 there corresponds an x E X such that
Ilxll s;{)-lilyll and Ily-Axil