Articles of functional analysis

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 : $$f_n((x_j))=\sum_{k=1}^{n}{\frac{x_k}{k}} , (x_j) \in \ell_2\,?$$

$\ker f$ is either dense or closed when $f$ is a linear functional on a normed linear space

Let $f$ be a linear functional on a normed linear space $X$. Prove that $\ker f$ is either dense or closed in $X$. Two possibilities can occur, i.e either $f$ is bounded or unbounded. If it is bounded then $f$ is continuous and hence $\ker f=f^{-1}(0)$ which is closed How to show when $f$ is […]

When does a dense subspace destine the weak topology?

Let $E$ be a locally convex space, let $E^{\prime}$ be its continuous dual space and let $F$ be a subspace of $E^{\prime}$ which is dense with respect to the strong topology on $E^{\prime}$ (i.e. the topology of uniform convergence on bounded subsets on $E$). Under what assumptions on $E$ is it possible to conclude that […]

Weak topology and the topology of pointwise convergence

I’m reading the definition of weak topology in Banach Algebra Techniques in Operator Theory by Douglas: According to an article about the product topology in Wikipedia, the product topology is also called topology of pointwise convergence. I’m confused with the underscored sentence. According to the answer by @Brian M. Scott to the question What is […]

Do Incomplete Normed Vector Spaces Whose Duals Are Reflexive Exist?

It is clear to me that if $X$ is a Banach space and its dual $X^*$ is reflexive, then $X$ is also reflexive (that is, the natural map between $X$ and its double dual $X^{**}$ is a surjective isometric isomorphism). However, I suspect that the conclusion fails to be true if we remove the completeness hypothesis, […]