Articles of functional analysis

Topological vector spaces vs. locally convex spaces

I’m taking a course on locally convex spaces and our lecturer mentioned that these form the most general collection of spaces on which one can still prove interesting theorems (like Hahn-Banach – which fails for the more general topological vector spaces) and which still have applications (exactly because interesting theorems hold for locally convex spaces). […]

Is this functional weakly lower semicontinuous?

Define $F\colon L^2([0,1]) \to {\mathbb R}$ by $$ F(R) = \int_0^1 \int_0^1 R(t) R(t’) \exp\left(-|t-t’| – \left|\int_t^{t’} R(s)\,ds\right|\right) \,dt\,dt’.$$ Is $F$ weakly lower semicontinous, that is, do we have $F(R) \leq \liminf_{n\to\infty}F(R_n)$ if $R_n$ converges weakly in $L^2$ to $R$? This is not a homework problem.

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: $$F_n:\Omega\to\mathbb{C}:\quad F_n\restriction_\mathbb{R}=f_n$$ Does their uniform limit have an analytic continuation, too? $$F:\Omega\to\mathbb{C}:\quad F\restriction_\mathbb{R}=f\quad(f_n\stackrel{\infty}{\to}f)$$ (By uniform boundedness this seems very likely; but really?) Application An almost modular state is modular: $$A\in\mathcal{A}^\omega:\quad\omega(\sigma^t[A]B)=\omega(B\sigma^{t+i\beta}[A])\quad(B\in\mathcal{A})$$ (Supposed that entire elements are […]

Pointwise limit of convolution

Suppose $\omega$ is the standard mollifier in $\mathbb R$. Then, let $\omega_{\epsilon} (x):= \frac{1}{\epsilon} \omega \left(\frac{x}{\epsilon}\right)$. For $0 < t_{1} < t_{2}$ the following result is to be proven: $$ \lim_{\epsilon \to 0} \int_{0}^{t} (\omega_{\epsilon}(s-t_1) – \omega_{\epsilon}(s-t_2))\,ds = \chi_{[t_1,t_2]}, $$ where $\chi_{[t_1,t_2]}$ is the characteristic function on the interval $[t_1, t_2]$. I couldn’t prove the […]

Goldstine's theorem

Let $X$ be a Banach space and suppose $X^{\prime\prime}=A\oplus B$, where $A$ and $B$ are infinite dimensional and closed. Is $\kappa(X)\cap A$ weak*-dense in $A$? $\kappa\colon X\to X^{\prime\prime}$ is the standard embedding.

Positive compact operator has unique square root.

Let H be a hilbert space and T be a compact positive operator so that by the spectral decomposition theorem, $T=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,e_{n}\rangle e_{n}$ where the $e_{n}$ are the eigenvectors of T and form an orthonormal basis. I have shown that T has a positive square root namely $S=\sum_{n=1}^{\infty}{\lambda}_{n}^{0.5}\langle x,e_{n}\rangle e_{n}$. I am trying to show […]

A finite dimensional normed space

I would like to find a short proof for the following theorems: Theorem 1. A normed space is finite dimensional iff all of its linear functional is continuous. Theorem 2. A normed space is finite dimensional iff its unit ball is compact. Thank you in advance.

Density of $C^\infty(\overline{\Omega})$ in $L^2(\Omega)$: can we find a bounded sequence approximating $a \in L^2(\Omega)$

Let $a \in L^2(\Omega)$ (bounded $\Omega$) with $0 \leq a(x) \leq C$ a.e. We know $C^\infty(\overline{\Omega})$ is dense in $L^2(\Omega)$, so there exist smooth functions $a_n \to a$ in $L^2$. But can we find a sequence $a_n$ such that $0 \leq a_n(x) \leq C$ (a.e)? I think so. Because if $a_n \to a$ in $L^2$ […]

Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle

If we have a Lissajous curve given by the parametric equations $$x(t)=A\sin(\omega_x t)$$ $$y(t)=B\sin(\omega_y t + \delta),$$ how can we show that the curve is dense in the rectangle of sides $A,B$ if and only if $\omega_x$ and $\omega_y$ are incommensurate (i.e. their ratio is irrational)? What I tried: Suppose it is not dense, so […]

Weak and strong convergence of sequence of linear functionals

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