Page 225 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes If x is any vector in B we can by the Archimedean property of real numbers find a positive
integer n such that n > x , i.e., x S ,
n
therefore


B = S n
n 1
and since t is onto, we have
B = T (B)


= T S n
n 1


= T (S )
n
n 1
Now B being complete, Baire’s theorem implies that some T S possesses an interior point
n 0
Z . This in turn yields a point y T S such that y is also an interior point of T S .
0 0 n 0 0 n 0
Further, maps j : B B and g : B B
defined respectively by j (y) = y = y – y and g (y) = 2 n y
0 0
where n is a non-zero scalars, are homeomorphisms as shown below f is one-to-one and onto.
o
–1
To show f, f are continuous, let y B and y y in B.
n n
Then f (y ) = y – y y – y = f (y)
n n 0 0
–1
–1
and f (y ) = y + y y + y = f (y)
n n 0 0
Hence f and f are both continuous so that is a homeomorphism.
–1
Similarly g : B B : g (x) = 2n y is a homeomorphism for, g is one-to-one, onto and bicontinuous
0
for n 0.
0
Therefore we have
(i) f (y ) = 0 = origin in B is an interior point of f T S .
0 n
(ii) f T S = f T S
n 0
n 0
= T S y
n 0 0

T S  y T S
2 n 0 0 n 0

(iii) T S = T 2 S 2n T S
2 n 0 n 0 1 0 1
= g (T(S )) g (T(S ))
1
1
= 2 T(S )
n 0 1

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