Page 265 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes So, by Pythagorean theorem, we have
2
2
(x – x ) + (y – y ) = x – x + y – y .
2
m n m n m n m n
But (x – x ) + (y – y ) = z – z so that
m n m n m n
2
z – z 2 = x – x + y – y 2 … (1)
m n m n m n
Since (z ) is a convergent sequence in H, it is a Cauchy sequence in H.
n
Hence z – z 2 0 as m, n … (2)
m n
Using (2) in (1), we see that
x – x 0 and y – y 0
2
2
m n m n
So that (x ) and (y ) are Cauchy sequence in M and N.
n n
Since H is complete and M and N are closed subspace of a complete space H, M and N are
complete.
Hence, the Cauchy sequence (x ) in M converges to x in M and the Cauchy sequence (y ) in N
n n
converges to y in N.
Now z = lim z = lim (x + y )
n n n
= lim x + lim y
n n
But lim x + lim y = x + y M + N
n n
Thus, z = x + y M + N
M + N is closed.

24.1.5 The Orthogonal Decomposition Theorem or Projection Theorem

Theorem 5: If M is a closed linear subspace of a Hilbert space H, then H = M M .

Proof: If M is a subspace of a Hilbert space H, then we know that M M = {0}.
Therefore in order to show that
H = M M , we need to verify that

H = M + M .
Since M and M are closed subspace of H, M + M is also a closed subspace of H by theorem 4.
Let us take N = M + M and show that N = H.
From the definition of N, we get M N and M N. Hence by theorem (1), we have
N M and N M .

Hence N M M = {0}.
N = {0}
N = {0} = H … (1)

Since N = M + M is a closed subspace of H, we have by theorem (3),
N = N … (2)
From (1) and (2), we have
N = M + M = H.

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