Page 279 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
(ii) In the Hilbert space  , the set {e , e , …, e , …} where e is a sequence with 1 in the
2
1
n
2
n
n place and O’s elsewhere is an orthonormal set.
th
25.3 Keywords
Orthonormal Sets: A non-empty subset { e } of a Hilbert space H is said to be an orthonormal set
i
if
(i) i j e e, equivalently i j (e , e) = 0
i j i j
(ii) e = 1 or (e , e) = 1 for every i.
i i j
Unit Vector or Normal Vector: Let H be a Hilbert space. If x H is such that x = 1, i.e. (x, x) =
1, then x is said to be a unit vector or normal vector.

25.4 Review Questions

1. Let {e , e , …, e } be a finite orthonormal set in a Hilbert space H, and let x be a vector in H.
1 2 n
n
If , , …, are arbitrary scalars, show that x e attains its minimum value
1 2 n i i
i 1
= (x, e ) for each i.
i i
2. Prove that a Hilbert space H is separable every orthonormal set in H is countable.

25.5 Further Readings

Book Sheldon Axler, Linear Algebra Done Right (2nd ed.), Berlin, New York (1997).

Online links www.mth.kcl.ac.uk/~jerdos/op/w3.pdf
mathworld.wolfram.com
www.utdallas.edu/dept/abp/PDF_files

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