In mathematics, weak convergence in a Hilbert space is the 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 {\displaystyle H} is said to converge weakly to a point x {\displaystyle x} in H {\displaystyle 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 {\displaystyle y} in H {\displaystyle 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 (that is, if it converges in norm), 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 infinite-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 → x {\displaystyle x_{n}\to x} weakly and ‖ x n ‖ → ‖ x ‖ {\displaystyle \lVert x_{n}\rVert \to \lVert x\rVert } , then x n → x {\displaystyle x_{n}\to 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 weak and strong convergence are equivalent.
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
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