In mathematics, weak convergence in a Hilbert
space is convergence of a sequence of points
in the weak topology.
== Definition ==
A sequence of points
(
x
n
)
{\displaystyle (x_{n})}
in a Hilbert space H is said to converge weakly
to a point x in H if
⟨
x
n
,
y
⟩
→
⟨
x
,
y
⟩
{\displaystyle \langle x_{n},y\rangle \to
\langle x,y\rangle }
for all y in H. Here,
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle
}
is understood to be the inner product on the
Hilbert space. The notation
x
n
⇀
x
{\displaystyle x_{n}\rightharpoonup x}
is sometimes used to denote this kind of convergence.
== Properties ==
If a sequence converges strongly, then it
converges weakly as well.
Since every closed and bounded set is weakly
relatively compact (its closure in the weak
topology is compact), every bounded sequence
x
n
{\displaystyle x_{n}}
in a Hilbert space H contains a weakly convergent
subsequence. Note that closed and bounded
sets are not in general weakly compact in
Hilbert spaces (consider the set consisting
of an orthonormal basis in an infinitely dimensional
Hilbert space which is closed and bounded
but not weakly compact since it doesn't contain
0). However, bounded and weakly closed sets
are weakly compact so as a consequence every
convex bounded closed set is weakly compact.
As a consequence of the principle of uniform
boundedness, every weakly convergent sequence
is bounded.
The norm is (sequentially) weakly lower-semicontinuous:
if
x
n
{\displaystyle x_{n}}
converges weakly to x, then
‖
x
‖
≤
lim inf
n
→
∞
‖
x
n
‖
,
{\displaystyle \Vert x\Vert \leq \liminf _{n\to
\infty }\Vert x_{n}\Vert ,}
and this inequality is strict whenever the
convergence is not strong. For example, infinite
orthonormal sequences converge weakly to zero,
as demonstrated below.If
x
n
{\displaystyle x_{n}}
converges weakly to
x
{\displaystyle x}
and we have the additional assumption that
‖
x
n
‖
→
‖
x
‖
{\displaystyle \lVert x_{n}\rVert \to \lVert
x\rVert }
, then
x
n
{\displaystyle x_{n}}
converges to
x
{\displaystyle x}
strongly:
⟨
x
−
x
n
,
x
−
x
n
⟩
=
⟨
x
,
x
⟩
+
⟨
x
n
,
x
n
⟩
−
⟨
x
n
,
x
⟩
−
⟨
x
,
x
n
⟩
→
0.
{\displaystyle \langle x-x_{n},x-x_{n}\rangle
=\langle x,x\rangle +\langle x_{n},x_{n}\rangle
-\langle x_{n},x\rangle -\langle x,x_{n}\rangle
\rightarrow 0.}
If the Hilbert space is finite-dimensional,
i.e. a Euclidean space, then the concepts
of weak convergence and strong convergence
are the same.
=== Example ===
The Hilbert space
L
2
[
0
,
2
π
]
{\displaystyle L^{2}[0,2\pi ]}
is the space of the square-integrable functions
on the interval
[
0
,
2
π
]
{\displaystyle [0,2\pi ]}
equipped with the inner product defined by
⟨
f
,
g
⟩
=
∫
0
2
π
f
(
x
)
⋅
g
(
x
)
d
x
,
{\displaystyle \langle f,g\rangle =\int _{0}^{2\pi
}f(x)\cdot g(x)\,dx,}
(see Lp space). The sequence of functions
f
1
,
f
2
,
…
{\displaystyle f_{1},f_{2},\ldots }
defined by
f
n
(
x
)
=
sin
⁡
(
n
x
)
{\displaystyle f_{n}(x)=\sin(nx)}
converges weakly to the zero function in
L
2
[
0
,
2
π
]
{\displaystyle L^{2}[0,2\pi ]}
, as the integral
∫
0
2
π
sin
⁡
(
n
x
)
⋅
g
(
x
)
d
x
.
{\displaystyle \int _{0}^{2\pi }\sin(nx)\cdot
g(x)\,dx.}
tends to zero for any square-integrable function
g
{\displaystyle g}
on
[
0
,
2
π
]
{\displaystyle [0,2\pi ]}
when
n
{\displaystyle n}
goes to infinity, which is by Riemann–Lebesgue
lemma, i.e.
⟨
f
n
,
g
⟩
→
⟨
0
,
g
⟩
=
0.
{\displaystyle \langle f_{n},g\rangle \to
\langle 0,g\rangle =0.}
Although
f
n
{\displaystyle f_{n}}
has an increasing number of 0's in
[
0
,
2
π
]
{\displaystyle [0,2\pi ]}
as
n
{\displaystyle n}
goes to infinity, it is of course not equal
to the zero function for any
n
{\displaystyle n}
. Note that
f
n
{\displaystyle f_{n}}
does not converge to 0 in the
L
∞
{\displaystyle L_{\infty }}
or
L
2
{\displaystyle L_{2}}
norms. This dissimilarity is one of the reasons
why this type of convergence is considered
to be "weak."
=== Weak convergence of orthonormal sequences
===
Consider a sequence
e
n
{\displaystyle e_{n}}
which was constructed to be orthonormal, that
is,
⟨
e
n
,
e
m
⟩
=
δ
m
n
{\displaystyle \langle e_{n},e_{m}\rangle
=\delta _{mn}}
where
δ
m
n
{\displaystyle \delta _{mn}}
equals one if m = n and zero otherwise. We
claim that if the sequence is infinite, then
it converges weakly to zero. A simple proof
is as follows. For x ∈ H, we have
∑
n
|
⟨
e
n
,
x
⟩
|
2
≤
‖
x
‖
2
{\displaystyle \sum _{n}|\langle e_{n},x\rangle
|^{2}\leq \|x\|^{2}}
(Bessel's inequality)where equality holds
when {en} is a Hilbert space basis. Therefore
|
⟨
e
n
,
x
⟩
|
2
→
0
{\displaystyle |\langle e_{n},x\rangle |^{2}\rightarrow
0}
(since the series above converges, its corresponding
sequence must go to zero)i.e.
⟨
e
n
,
x
⟩
→
0.
{\displaystyle \langle e_{n},x\rangle \rightarrow
0.}
== Banach–Saks theorem ==
The Banach–Saks theorem states that every
bounded sequence
x
n
{\displaystyle x_{n}}
contains a subsequence
x
n
k
{\displaystyle x_{n_{k}}}
and a point x such that
1
N
∑
k
=
1
N
x
n
k
{\displaystyle {\frac {1}{N}}\sum _{k=1}^{N}x_{n_{k}}}
converges strongly to x as N goes to infinity.
== Generalizations ==
The definition of weak convergence can be
extended to Banach spaces. A sequence of points
(
x
n
)
{\displaystyle (x_{n})}
in a Banach space B is said to converge weakly
to a point x in B if
f
(
x
n
)
→
f
(
x
)
{\displaystyle f(x_{n})\to f(x)}
for any bounded linear functional
f
{\displaystyle f}
defined on
B
{\displaystyle B}
, that is, for any
f
{\displaystyle f}
in the dual space
B
′
{\displaystyle B'}
. If
B
{\displaystyle B}
is an Lp space on
Ω
{\displaystyle \Omega }
, and
p
<
∞
{\displaystyle p<\infty }
then, any such
f
{\displaystyle f}
has the form
f
(
x
)
=
∫
Ω
x
y
d
μ
{\displaystyle f(x)=\int _{\Omega }x\,y\,d\mu
}
For some
y
∈
L
q
(
B
)
{\displaystyle y\in \,L^{q}(B)}
where
1
p
+
1
q
=
1
{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1}
and
μ
{\displaystyle \mu }
is the measure on
Ω
{\displaystyle \Omega }
.
In the case where
B
{\displaystyle B}
is a Hilbert space, then, by the Riesz representation
theorem,
f
(
⋅
)
=
⟨
⋅
,
y
⟩
{\displaystyle f(\cdot )=\langle \cdot ,y\rangle
}
for some
y
{\displaystyle y}
in
B
{\displaystyle B}
, so one obtains the Hilbert space definition
of weak convergence.
