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.

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 […]

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.

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 $L^2$ […]

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 […]

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

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 […]

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 […]

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 […]

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, […]

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