Articles of functional analysis

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 […]

Is the unit sphere in $(C, \| \cdot\|_1)$ compact?

Consider the normed space $(C[0,1], \| \cdot\|_1)$ where $C[0,1]=\{f:[0,1] \to \Bbb R : f$ is continuous$\}$ and $\|f\|_1 = \int_0^1|f(t)|dt$. I’m trying to find out if the unit sphere $S=\{f \in C[0,1] : \| f \|_1 = 1\}$ is compact or not. To prove it’s not compact (I don’t know if that’s true or not) […]

Proving that a compact subset of a Hausdorff space is closed

I am having trouble understanding the answers here. I am trying to prove that a compact subset of a Hausdorff space is closed. Following the proof is difficult, perhaps because Brian reused letters for different things(although I get they are arbitrary, I can’t follow it.) The second answer uses nets and filters, which I don’t […]

Weak derivative in Sobolev spaces

A function $u: \Bbb R \longrightarrow \Bbb R$ is weakly differentiable with weak derivative $v$ if there exists a function $v: \Bbb R \longrightarrow \Bbb R$ such that $$\int_{-\infty}^\infty u \phi’ ~dx = – \int_{-\infty}^\infty v \phi ~dx$$ for all smooth functions $\phi: \Bbb R \longrightarrow \Bbb R$ that vanish outside some bounded set. Functions […]

$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin’s (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq a<b\leq\infty$, is not equal to 0 [i.e. $\forall x\in(a,b)\quad f(x)\ne 0$, as Daniel, whom I deeply thank, explains in his answer] and satisfy the condition $|f(x)|\leq Ce^{-\delta|x|}$ with $\delta>0$, then […]

uniqueness of Hahn-Banach extension for convex dual spaces

Let $X’$ be strict convex, i.e. for all $x_1′,x_2’\in X’$ with $\|x_1’\|_{X’}=\|x_2’\|_{X’}=1$ the implication $$\left\|\frac{x_1’+x_2′}{2}\right\|=1\Rightarrow x_1’=x_2’$$ holds. In this case the Hahn-Banach-extension is unique. I am trying to figure out how I can show this. The Hahn-Banach theorem says that for a subspace $U\subset X$ of a normed space $X$, there exists an extension $x’\in […]

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u’v_0’+uv_0), \forall u \in H^1(0,1)$. Further Show that $v_0$ is the solution of some differential equation with appropriate boundary conditions. Compute $v_0$ explicitly. Let $f: H^1(0,1) \to \mathbb{R}$ be defined by $f(u)=u(0)$. Then I showed that $f$ is linear and continuous. Hence there exists […]

Product and sum of positive operators is positive

I want to show that for $S,T\in B(H)$ bounded operators on Hilbertspace with $S\geq 0,T\geq 0$ and $ST=TS$, we have $S+T\geq 0$, and $ST\geq 0$. $T\geq 0$ means $(Tx,x)\geq 0$. To me it seems that $((S+T)x,x) = (Sx,x)+(Tx,x)\geq 0$. But for $ST$ i like some help. There is a theorem in Rudin (12.32) that says […]

Reflexive but not separable space

I’m trying to find an example of normed vector space that is reflexive but not separable. (Separable but not reflexive is easy, for example $L^1$).

Weak convergence and weak convergence of time derivatives

I am working in $H^1(S^1)$, the space of absolutely continuous $2\pi$-periodic functions $\mathbb R\to\mathbb R^{2n}$ wih square integrable derivwtives. I have a sequence $z_j$ (for the record, it comes from minimizing a functional, in the middle of Hofer-Zehnder’s proof that a Hamiltonian field has a periodic orbit on a strictly convex compact regular energy surface) […]