Pre-Hilbest Spaces
1) Inner product and associated norm
:
#
ef Vuestor Space #
An inner product (or scalar product)
:
over
2
. , ) VxV) FeEV
>
,
(e vo
(positivity
> v=
Or
(definiteness)
=
,
0
iii) F ,
2 , 23
Sen enters)
,
V is a
map
# such that
:
:
ii) (v , w )
on
EV ,
=
12 5) + <m 3)
,
,
Fadditivity with respect 2nd slot)
iv) FX EIF Funiz EV
,
(en Xez)
,
,
Chorogeneity wresp to 2ndslot)
=
.
1) FeeEV (Uniwa)
,
(V ) ()
,
A
.,
.
isan inner
=
over
,
id if F=R
[Na,e)omconji
-
Conjugate symmetry
product space
finite dim pretilbest space
X (e we)
,
or
pre-Hilbert space
R is called Euclidean
A
finite dim pretilbest space
Remark
:
X12 3)
=
+
Xez
2)
,
[vz
=
,
=
(m
+
,
,
2+
is
called Hermitian
Stinearity)
(en + n) + X(v1 23)
,
,
,
↑ ea Xez)
+
K
Fr Ve us EV XXEF
iii)
by
and iv]
Se
over
=
,
,
Xe2)
(0z , ) + X (23 527
.
(vi) + Eve]
=
(e vs) + (vz
,
,
2) Canlineage
Exemples
:
ess) y (yn yzyz] EIR3
(1
(e y) xyn xzyz + ezy
er =
,
er ,
2) r (err
=
-
-
-
N
,
,
es =
=
. . .
,
---
[x]C
: (s
matrices
of
y) +XY
Y [y]e
Hermitian
=
:
in terms
Jes ren) y (yn
(e y) =
,
,
xn) y (yn X yn) ERI
Jes y) = sig
3)
=
+
=
,
,
=
=
,
=
--
-
conjuguate
yn)EC"
J
=
sy : in terms of matrics (x y
,
tXX
nXY
Theorem
JU
S
.
(Cauchy Schwartz inequality)
:
-
1
.
)) pre-fillet
Ye yev
,
/(e y)/2[s e) < y y)
,
,
equality if and of
then
# if <es y)
-and
=
,
-
otherwise
colinear
only
re ,
0
we are
,
y)
=
0
,
,
,
y
done
FX EIR
and write
Kes y)/
,
0x(0x + Xy
=
,
xy)
Ou +
O (n x) +
,
~
=
0(0x + Xy x) +X(0u + Xy ,
=
,
Xo(y m)
,
+
x(x y) +
,
1
y)
X (y y)
,
Oral
Observe that
Kee y7/2
,
&
-
I
(u e) [us
y)
<y u) =Kee
,
,
,
,
Jes y)
,
=
on (n n) +
,
-
Yo
=
Ke y)/
y)/
Kno yi/
,
,
Ke y)/
Hence YX ERR b
-
Kee e)
,
=
,
a
y) deg2 polynomial
2x +
positive coeff
with
IR
.
↳ *
=>
=>
=>
X/ <
y)/24(e x) <y y)
e,
,
,
Kee /2) (e) (y y) => Kn %)/XMeT
,
the
.
.
becomes an equality
inequality (iff
<y
,
0
y)/faq z)
=
there's a root)
Fand only if EXER such that
(Ox +Xy 0x + Xy)
,
Ou +
E
definiteness
#
s,
thm
y
are
Xy
0u=
-y
↑
colinear.
Minkowski
:
=
0
=
nonzerolbyassumption]
inequality.
)) poe-Hilbertspace
Feey EV Lety y] - Ses us] Seeing
with
equality if and only if EX EIR + /y =
&
) (y y)
roof <sety
y) (ne x) (x y)+
(
,
C
...
+
, e
,
+
,
ne
:
,
x+
=
+
,
+
,
-
=
[esig]
-
(re y y) - (n n) +2)(x y))+ (y y)[Rakee y))
> (ee e) +2
use) < y y)
<y y)
+
(x y
+
,
,x
+
,
-
, x+
,
,
y)())(e
,
,
u) +
,
(y y))2
,
+
,
< n y))
,
,
Th The mapN
-
:
is a norm on
si
V
Proof Exercise
:
Exples
:
-
Jesy) = Digi
1) e yeR
.
,
Hell
2) x ye"(r y)
,
,
rely on
= gi
Ilvell = :13 =
N
Ifll Saifitdt
=
4)
.
,
u,
012 (N
1) all
=
&
the
verify
eroded
all
on
fig eco (lab] 4) <fig
3)
+
F
g(tdt
nam2
Su ,Y
=n
=ou
↑
,
1)
·
why
is
this infinite scus
well def
.
Sexercise)