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Weak convergence (Hilbert space) - Wikipedia

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Type of convergence in Hilbert spaces

In mathematics, weak convergence in a Hilbert space is the convergence of a sequence of points in the weak topology.

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

lim n → ∞ ⟨ x n , y ⟩ = ⟨ x , y ⟩ {\displaystyle \lim _{n\to \infty }\langle x_{n},y\rangle =\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.^[1]

‖ 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.

⟨ 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 weak and strong convergence are equivalent.

The first three functions in the sequence f n ( x ) = sin ⁡ ( n x ) {\displaystyle f_{n}(x)=\sin(nx)} on [ 0 , 2 π ] {\displaystyle [0,2\pi ]} . As n → ∞ {\displaystyle n\rightarrow \infty } f n {\displaystyle f_{n}} converges weakly to f = 0 {\displaystyle f=0} .

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 L^p 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[edit]

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 {e_n} 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[edit]

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.

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 ( Ω ) {\displaystyle y\in \,L^{q}(\Omega )} , where μ {\displaystyle \mu } is the measure on Ω {\displaystyle \Omega } and 1 p + 1 q = 1 {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1} are conjugate indices.

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.