Articles of functional analysis

Does that fact that the dual map is zero imply that the map is zero?

Let $X$ and $Y$ be vector spaces over a field $\mathbb{F}$ and let $f:X\to Y$ be a linear map. If the dual map $f^*: Y^*\to X^∗:y^*\mapsto y^*\circ f$ is the zero map, is the original map $f$ the zero map too?

mononotone and pseudomonotone operators in current research

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone operators in mathematics or applied science in current research. I am aware of the use in proving the existence of solutions to […]

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I’m reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $Q_1$-Wiener process on a “larger” Hilbert space. I’ve got the feeling that $Q_1$ is actually a really “simple” operator (like the identity or […]

Representing the tensor product of two algebras as bounded operators on a Hilbert space.

Hi Math StackExchange, Let $A$ be a commutative, infinite dimensional, unital, *-algebra represented by bounded operators on a Hilbert space $H_A$. Next let $B$ be a finite non-commutative *-algebra represented on a Hilbert space $H_B$. Is it true in general that the product $A\otimes B$ is represented by $\textbf{bounded}$ operators on $H_A\otimes H_B$? How could […]

SOT limit of Self-adjoint operators is self-adjoint?

Say $T_n$ is a sequence of self-adjoint operators on a Hilbert space and converges in the strong operator topology to $\mathcal{T}$, must $\mathcal{T}$ be self-adjoint? Since $T_nx$ converges to $\mathcal{T}x$ in norm, it converges weakly, and so I figured that $\langle\mathcal{T}x,x\rangle-\langle x,\mathcal{T}x\rangle=\lim_{n\to \infty}\langle T_nx, x\rangle-\langle x, T_nx\rangle=0,$ which does the job, but there’s a chance […]

Where does the iteration of the exponential map switch from one fixpoint to the 3-periodic fixpoint cycle?

In the answering of another question in MSE I’ve dealt with the iteration of $x=b^x$ where the base $b=i$. If I reversed that iteration $x=log(x)/log(i)$ then I run into a cycle of 3 periodic fixpoints. Another complex base on the unit circle however, $b = \sqrt{0.5}(1+i)$ gives only one fixpoint. I asked myself, whether there […]

The Gelfand transformation on $\ell^1(\mathbb Z)$ is not isometric. Do you have an example?

I am looking for an element $(a_n)_{n\in\mathbb Z}\in\ell^1(\mathbb Z)$ with the property $$ \lVert(a_n)\rVert_{\ell^1(\mathbb Z)} > \lVert\sum_{n\in\mathbb Z}a_n z^n\rVert_\infty, $$ where the norm on the right hand side denotes the sup-norm on $\mathcal C(\mathbb T)$ ($\mathbb T$ is the 1-Torus). Motivation: I want to prove that the Gelfand transformation on $\ell^1(\mathbb Z)$ is not isometric.

How to show projection of $L^2$ function converges to that $L^2$ function

My teacher said that if $P_n f = \sum_{j=0}^n(f,w_j)w_j$, where $w_j$ is orthonormal basis of $L^2$, then $|P_n f- f|_{L^2} \to 0$ for $f \in L^2$. How do I prove this? I thought $$|P_nf – f| = |\sum_{j=0}^n(f,w_j)w_j – \sum_{j=0}^\infty(f,w_j)w_j| = |\sum_{j={n+1}}^\infty(f,w_j)w_j| \leq \sum_{j={n+1}}^\infty|f|$$ where the last equality is by Cauchy Schwarz, but this equals […]

Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other…

Are polynomials dense in $C^k\left(\bar{B}\right)$?

Let $\bar{B}$ be the closed unit ball in $\mathbb{R}^n$, $C^k\left(\bar{B}\right)$ the Banach space of all real function defined on $\bar{B}$ with continuous derivatives up to order $k$, with norm $$\Vert f \Vert = \sum_{h\le k} \Vert \partial_{i_1}\dots \partial_{i_h}f\Vert_\infty$$ Are polynomials dense in $C^k\left(\bar{B}\right)$? Can you give me references about that subject? Thanks a lot…