:: by Hiroshi Yamazaki , Yasumasa Suzuki , Takao Inou\'e and Yasunari Shidama

::

:: Received April 17, 2003

:: Copyright (c) 2003-2021 Association of Mizar Users

Lm1: for x, y, z, e being Real st |.(x - y).| < e / 2 & |.(y - z).| < e / 2 holds

|.(x - z).| < e

proof end;

Lm2: for p being Real st p > 0 holds

ex k being Nat st 1 / (k + 1) < p

proof end;

Lm3: for p being Real

for m being Nat st p > 0 holds

ex i being Nat st

( 1 / (i + 1) < p & i >= m )

proof end;

theorem Th1: :: BHSP_7:1

for X being RealUnitarySpace

for Y being Subset of X

for L being Functional of X holds

( Y is_summable_set_by L iff for e being Real st 0 < e holds

ex Y0 being finite Subset of X st

( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds

|.(setopfunc (Y1, the carrier of X,REAL,L,addreal)).| < e ) ) )

for Y being Subset of X

for L being Functional of X holds

( Y is_summable_set_by L iff for e being Real st 0 < e holds

ex Y0 being finite Subset of X st

( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds

|.(setopfunc (Y1, the carrier of X,REAL,L,addreal)).| < e ) ) )

proof end;

theorem Th2: :: BHSP_7:2

for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds

for S being finite OrthogonalFamily of X st not S is empty holds

for I being Function of the carrier of X, the carrier of X st S c= dom I & ( for y being Point of X st y in S holds

I . y = y ) holds

for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds

H . y = y .|. y ) holds

(setopfunc (S, the carrier of X, the carrier of X,I, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,I, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)

for S being finite OrthogonalFamily of X st not S is empty holds

for I being Function of the carrier of X, the carrier of X st S c= dom I & ( for y being Point of X st y in S holds

I . y = y ) holds

for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds

H . y = y .|. y ) holds

(setopfunc (S, the carrier of X, the carrier of X,I, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,I, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)

proof end;

theorem Th3: :: BHSP_7:3

for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds

for S being finite OrthogonalFamily of X st not S is empty holds

for H being Functional of X st S c= dom H & ( for x being Point of X st x in S holds

H . x = x .|. x ) holds

(setsum S) .|. (setsum S) = setopfunc (S, the carrier of X,REAL,H,addreal)

for S being finite OrthogonalFamily of X st not S is empty holds

for H being Functional of X st S c= dom H & ( for x being Point of X st x in S holds

H . x = x .|. x ) holds

(setsum S) .|. (setsum S) = setopfunc (S, the carrier of X,REAL,H,addreal)

proof end;

theorem Th4: :: BHSP_7:4

for X being RealUnitarySpace

for Y being OrthogonalFamily of X

for Z being Subset of X st Z is Subset of Y holds

Z is OrthogonalFamily of X

for Y being OrthogonalFamily of X

for Z being Subset of X st Z is Subset of Y holds

Z is OrthogonalFamily of X

proof end;

theorem Th5: :: BHSP_7:5

for X being RealUnitarySpace

for Y being OrthonormalFamily of X

for Z being Subset of X st Z is Subset of Y holds

Z is OrthonormalFamily of X

for Y being OrthonormalFamily of X

for Z being Subset of X st Z is Subset of Y holds

Z is OrthonormalFamily of X

proof end;

theorem Th6: :: BHSP_7:6

for X being RealHilbertSpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds

for S being OrthonormalFamily of X

for H being Functional of X st S c= dom H & ( for x being Point of X st x in S holds

H . x = x .|. x ) holds

( S is summable_set iff S is_summable_set_by H )

for S being OrthonormalFamily of X

for H being Functional of X st S c= dom H & ( for x being Point of X st x in S holds

H . x = x .|. x ) holds

( S is summable_set iff S is_summable_set_by H )

proof end;

theorem Th7: :: BHSP_7:7

for X being RealUnitarySpace

for S being Subset of X st S is summable_set holds

for e being Real st 0 < e holds

ex Y0 being finite Subset of X st

( not Y0 is empty & Y0 c= S & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= S holds

|.(((sum S) .|. (sum S)) - ((setsum Y1) .|. (setsum Y1))).| < e ) )

for S being Subset of X st S is summable_set holds

for e being Real st 0 < e holds

ex Y0 being finite Subset of X st

( not Y0 is empty & Y0 c= S & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= S holds

|.(((sum S) .|. (sum S)) - ((setsum Y1) .|. (setsum Y1))).| < e ) )

proof end;

Lm4: for p1, p2 being Real st ( for e being Real st 0 < e holds

|.(p1 - p2).| < e ) holds

p1 = p2

proof end;

theorem :: BHSP_7:8

for X being RealHilbertSpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds

for S being OrthonormalFamily of X

for H being Functional of X st S c= dom H & ( for x being Point of X st x in S holds

H . x = x .|. x ) & S is summable_set holds

(sum S) .|. (sum S) = sum_byfunc (S,H)

for S being OrthonormalFamily of X

for H being Functional of X st S c= dom H & ( for x being Point of X st x in S holds

H . x = x .|. x ) & S is summable_set holds

(sum S) .|. (sum S) = sum_byfunc (S,H)

proof end;