Page 227 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Thus (s ) is a Cauchy sequence in B and since B is complete, a vector x B such that
n
s x and therefore
n
x = lim s n lim s n 2 3 ,
n n
i.e., x S .
3
It now follows by the continuity of T that

T (x) = T lim s n
n

= lim T(s )
n
n
= lim (y 1 y 2  y )
n
n
= y
Hence y T (S )
3
Thus y S y T (S ). Accordingly
3
S T (S )
3
This completes the proof of the lemma.

Note If B and B are Banach spaces, the symbol S (x; r) and S (x; r) will be used to denote
open spheres with centre x and radius r in B and B respectively. Also S and S will denote
r r
these spheres when the centre is the origin. It is easy to see that
S (x; r) = x + S and S = r S
r r 1
For, we have
y S (x; r) y – x < r
z < r and y – x = z, z S
r
y = x + z and z < r

y x + S
r
Thus S (x; r) = x + S
r
x
and S = {x : x < r} = x : 1
r r

= {r . y y < 1}
= r S
1
Thus S = r S
r 1
Now we prove an important lemma which is key to the proof of the open mapping theorem.

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