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, […]

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 […]

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?

“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? 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 $T=T^*$?

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 […]

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

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.

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 $\Phi$ […]

Intereting Posts

Roots of unity and a system of equations by Ramanujan
Infinite sums of reciprocal power: $\sum\frac1{n^{2}}$ over odd integers
minimal primes of a homogeneous ideal are homogeneous
$(x_1-a_1, x_2-a_2)$ is a maximal ideal of $K$
Matrices that are not diagonal or triangular, whose eigenvalues are the diagonal elements
Riemann Integral Upper vs Lower Estimate. Inf vs sup?
Differentiate a recurrence relation
Matrices (Hermitian and Unitary)
Solving $2^x \equiv x \pmod {11}$
Choosing two random numbers in $(0,1)$ what is the probability that sum of them is more than $1$?
Why is Peano arithmetic undecidable?
Is there a solution to $a^4+(a+d)^4+(a+2d)^4+(a+3d)^4+\dots = z^4$?
Proving $\frac2\pi x \le \sin x \le x$ for $x\in $
Integral $\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}$
Integral of gaussian distribution divided by square root of x