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Topic: RIESZ-FISCHER THEOREM (matrix form and proof)
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Alexander Abian

Posts: 424
Registered: 12/8/04
RIESZ-FISCHER THEOREM (matrix form and proof)
Posted: Jul 15, 1996 10:02 AM
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Dear Emma,

You asked me to reveal the gist of the statement and the proof of the
Riesz-Fisher Theorem. Here tyey are.

Everything is in the Hilbert space H of the square summable real sequences.

Let M be an infinite square matrix whose rows form an orthonormal
set of elements of H (the columns of H need not form an orthonormal
set, however, the norm of each column of M is necessarily less than or
equal to 1).

Let c be an element of H. Then the Riesz-Fischer Theorem can be stated
in matrix form as follows.

THEOREM. The equation

(1) Mx = c (x and c are written as column vectors)

has a solution x = k such that k is an element of H, the norm of
k is equal to the norm of c and the elements of c are the Fourier
coefficients (w.r.t the rows of M) of k.


PROOF. Let M' be the transpose of M. Clearly,

(2) MM' = the infinite unit matrix

Thus, from (2) it follows that

(3) (MM') c = c

But then with a little effort (since the product of infinite matrices
need not be associative, although for the special case (3) it is) from
(3) we obtain

(4) M(M'c) = c

But then it can be readily verified that (4) implies that

k = M'c

is the desired solution x = k mentioned above.

P.S. I did not give details since I did not want to obfuscate the issue.

Alexander Abian: Equivalence of Mass and Time
Albert Einstein: Equivalence of Mass and Energy
THERE WAS NO BIG BANG. THERE WAS A BIG SUCK INTO THE VOID OF SPACE
--

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TIME IS MASS. ABIAN MASS-TIME EQUIVALENCE FORMULA T=-(10^18)Log(1-m/Mo) SECONDS
ALTERING EARTH'S ORBIT AND TILT - STOPPING GLOBAL DISASTERS AND EPIDEMICS.
ALTERING THE SOLAR SYSTEM. REORBITING VENUS INTO A NEAR EARTH-LIKE ORBIT
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