Page 267 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes or |(z , y)| { y – 2} 0 … (2)
2
2
o
The above result is true for all real suppose that (z , y) 0. Then taking positive and so small
o
that y < 2, we see from (2) that |(z , y)| { y – 2} < 0.
2
2
2
o
This contradicts (2).
Hence we must have (z , y ) = 0 z y, y M.
o o o
z M.
o
This completes the proof of the theorem.
24.2 Summary
 Let H be a Hilbert space. If x, y H then x is said to be orthogonal to y, written as x y, if
(x, y) = 0.

 If x and y are any two orthogonal vectors in a Hilbert space H, then
2
2
2
x + y 2 = x – y = x + y .
 Two non-empty subsets S and S of a Hilbert space H are said to be orthogonal denoted by
1 2
S S , if x y for every x S and every y S .
1 2 1 2
 Let S be a non-empty subsets of a Hilbert space H. The orthogonal compliment of S,
written as S and is read as S perpendicular, is defined as
S = {x H : x y y S}
 The orthogonal decomposition theorem: If M is a closed linear subspace of a Hilbert space
H, then H = M M .

24.3 Keywords

Orthogonal Compliment: Let S be a non-empty subset of a Hilbert space H. The orthogonal
compliment of S, written as S and is read as S perpendicular, is defined as

S = {x H : x y y S}
Orthogonal Sets: A vector x is to be orthogonal to a non-empty subset S of a Hilbert space H,
denoted by x S if x y for every y in S.
Two non-empty subsets S and S of a Hilbert space H are said to be orthogonal denoted by S
1 2 1
S , if x y for every x S and every y S .
2 1 2
Orthogonal Vectors: Let H be a Hilbert space. If x, y H then x is said to be orthogonal to y,
written as x y, if (x, y) = 0.
Pythagorean Theorem: If x and y are any two orthogonal vectors in a Hilbert space H, then

2
2
2
x + y = x – y = x + y .
2
24.4 Review Questions
1. If S is a non-empty subset of a Hilbert space, show that S = S .
2. If M is a linear subspace of a Hilbert space, show that M is closed M = M .

3. If S is a non-empty subset of a Hilbert space H, show that the set of all linear combinations
of vectors in S is dense in H S = {0}.

260 LOVELY PROFESSIONAL UNIVERSITY