Articles of functional analysis

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, […]

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the localized Sobolev space $H_{s}^{loc}$ is the set of all distributions $f\in \mathcal{D’}(U)$ such that for every precompact open set $V$ with $\overline{V}\subset U$ there exists $g\in H_{s}$ such that $g=f$ on $V.$ Fact. A […]

Closed Operators: Spectrum

Given a Hilbert space $\mathcal{H}$. Consider operators: $$T:\mathcal{D}(T)\to\mathcal{H}$$ Suppose one has: $$T=\overline{T}=T^{**}$$ Then it may happen: $$\sigma(T)=\varnothing,\mathbb{C}$$ What are examples?

Infinite dimensional spaces other than functional spaces

“Functional analysis” is the study of infinite dimensional spaces equipped with inner product, norm, topology…etc. The most interesting spaces are the spaces of functions/operators and sequences. I don’t know if there’s “another kind” of infinite dimensional spaces other than space of functions/operators/sequences which is interesting.

Is a contraction idempotent operator self-adjoint?

Is a contraction idempotent operator self-adjoint? In the other words, if $T:H\to H$ is a bounded linear operator such that $||T||\leq1$ and $T^{2}=T$, can we conclude $T=T^*$?

Unconditional bases equivallent to permutations of basis elements.

On page 9 of The Handbook of the Geometry of Banach Spaces: Volume I I found the following: “A basis $\{x_n\}_{n=1}^{\infty}$ is said to be an unconditional basis provided that $\sum \alpha_n x_n$ converges unconditionally whenever it converges. This is equivalent to saying that every permutation of $\{x_n\}_{n=1}^{\infty}$ is also a basis.” How are those […]

Openness of linear mapping

Let $X$ be a topological vector space over the field $K$, where $K=\mathbb R$ or $K= \mathbb C$, and let $f:X\rightarrow K^n$ ($n \in \mathbb N$) be a linear and surjective functional. How to prove that $f$ is an open mapping? Thanks

Fourier transform and distibution beloning to S'

I need prove, that a distribution $$\langle F_f,\phi \rangle= p.v. \int\limits_{-1/2}^{1/2}\frac{\phi(t)}{t\cdot \ln{|t|}}\mathrm{d}t$$ belongs to $S^\prime$ (adjoint to Schwartz space) and I need find, Fourier transform of the distribution.

Isometry between $X^\ast/M^\perp$ and $M^\ast$.

Let $X$ be a normed linear space, $M \subset X$ be a subspace, $M^\perp = \{x^\ast \in X^\ast \mid x^\ast\big|_M = 0\}$ be the annihilator of $M$, $X^\ast$ the topological dual of $X$, and let’s define $\Phi: X^\ast/M^\perp \to M^*$ by $\Phi(x^\ast+M^\perp) = x^\ast\big|_M$. I have already proved that $M^\perp$ is closed, and that $\Phi$ […]