Page 271 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes We have y 2 = (y, y)

n n
= x (x, e ) e , x (x, e ) e
i i i i
i 1 i 1
n n
= (x,x) (x, e )(e , x) (x, e )(x, e )
j
i
i
j
i 1 j 1
n n
(x, e )(x, e ) (e , e )
i
j
i
i
i 1 j 1
n n n
2
= x (x, e )(x, e ) (x, e )(x, e ) (x, e )(x, e )
i
i
j
i
i
j
i 1 j 1 i 1
On summing with respect to j and remembering that (e , e) = 1, i = j and (e , e) = 0, i j
i j i j
n n n
2 2 2 2
= x x, e i x, e i (x, e )
i
i 1 i 1 i 1
n
2 2
= x (x, e )
i
i 1
n
2
2
Now y 0, therefore x – (x, e ) 0
2
i
i 1
n
2
(x, e ) x 2
i
i 1
result (1).
Further to prove result (2), we have for each j (1 j n),
n n
x (x, e ) e , e = (x, e ) (x, e ) e , e
i i j j i i j
i 1 i 1
n
= (x, e ) (x, e )(e , e )
j
i
j
i
i 1
= (x, e) – (x, e) [ (e , e) = 1, i j 0, i = j]
j j i j
= 0
n
Hence x (x, e ) e i e for each j.
j
i
i 1
This completes the proof of the theorem.
Note The result (1) is known as Bessel’s inequality for finite orthonormal sets.

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