Articles of weak convergence

weak convergence in $L^2$ / $C$ ==> pointwise convergence

I have a sequence of function $B_n \in C([0,1],R)$ and a $B \in C([0,1],R)$, such that: $\int_{[0,1]} B_n(x) f(x) dx \rightarrow \int_{[0,1]} B (x) f(x) dx$ for all $f \in C([0,1],R)$ which are bounded. Does this imply point wise convergence of $B_n(x) \rightarrow B(x)$ for all $x$? My starting idea: This convergence implies weak convergence […]

Product of weak/strong converging sequences

Let’s consider two sequences $u_n$ and $v_n$ such that $$u_n\to u\,,\,\,\,\rm{in}\,\,\,L^\infty(\mathbb{R}^n)$$ and $$v_n\rightharpoonup v\,,\,\,\,\rm{in}\,\,\,L^2(\mathbb{R}^n)$$ What can I say of the convergence of the product $u_nv_n$? In particoular I want that $u_nv_n$ converges weakly in $L^2_{loc}(\mathbb{R}^n)$. I procced as follows: since $u_n$ converges strongly in $L^\infty$ it converges strongly in $L^2_{loc}$ and so we have a […]

Weak and strong convergence of sequence of linear functionals

Is this sequence of linear functionals weakly (strongly) convergent : $$f_n((x_j))=\sum_{k=1}^{n}{\frac{x_k}{k}} , (x_j) \in \ell_2\,?$$

Characteristic function converges pointwise

Is there any sequence of probability distributions such that their characteristic functions converge pointwise, but the sequence of prob. distributions itself does not converge weakly? :S

Are the weak* and the sequential weak* closures the same?

I have a question that might be easy for the experts. Let $E$ be a Banach space (non-separable) and let $E’$ be its dual space. Suppose that $X\subset E’$ and assume that $X$ is separable with respect to the weak* topology. My question is the following: Are the sequential weak* closure and the weak* closure […]

Closed $\iff$ weakly closed subspace

On this link http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst_2004;task=show_msg;msg=1414.0001 is the argument that a linear subspace in a normed space is closed w.r.t. norm iff it is weakly closed. On the other hand, $c_0$ (sequences convergent to $0$) is a norm-closed linear subspace of $l_\infty$ (bounded sequences), but it is not weakly closed, since the base vectors $e_i$ are weakly […]

Weak topology is not metrizable: what's wrong with this proof?

Let $(X,\|\cdot\|)$ be an infinite-dimensional normed vector space. Suppose that the weak topology of $X$ is metrizable by a metric $d$. Denote by $B^d(x,r)$ the open balls with respect to $d$; they are therefore weakly open. We have that for every $n$ the ball $B^d(0,\frac{1}{n})$ contains a non-trivial subspace. We could then argue as follows: […]

Show a bounded linear operator is weakly sequentially continuous

Let $X$ and $Y$ be normed linear spaces, $T \in B(X,Y)$, and $\{x_{n}\}_{n=1}^{\infty}\subset X$. If $x_n \rightharpoonup x$, I need to prove that $T x_{n} \rightharpoonup T x$ in $Y$, where $\rightharpoonup$ denotes weak convergence.

In Hilbert space: $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$.

Assume that $H$ is a $\mathbb K$-Hilbert space, $(x_n)_{n \ge 1}$ a sequence in $H$ and $x ∈ H$. Show that $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$. I’m trying to prove this statement. The $\Rightarrow$ is basically clear, since a strongly convergent […]

Weak and strong convergence

I have the sequence $(v_n)\subset H^1_0(0,1)$ such that $v_n\rightharpoonup v $ (weakly) in $H^1_0(0,1)$ and $v_n\rightarrow v$ in $L^2(0,1)$ and $v_n\rightarrow v$ in $C^0(0,1)$ My question is why $$\int_0^1 v_n(x) (v_n(x)-v(x)) dx\rightarrow 0$$ I say that $\int_0^1 v_n (v_n-v) dx=\int_0^1 (v_n-v+v)(v_n-v) dx= \int_0^1 (v_n-v)^2 dx +\int_0^1 v(v_n-v) dx=$ $ ||v_n-v||^2_{L^2(0,1)} +\int_0^1 v(v_n-v) dx$ By the […]