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Let $E, F$ be Banach spaces. A continuous bilinear functional ${\langle \cdot\,, \cdot \rangle }: E \times F \to \mathbb{R}$ is called $E$-non-degenerate if $\langle x,y\rangle = 0$ for all $y \in F$ …

After my previous post I got curious about the following very simple question (which I don't seem to find the answer). Given a tempered distribution $K \in \mathcal{S}'(\mathbb{R}^{n_{1}+\cdots+n_{N}} …

Suppose $f: \mathcal{S}(\mathbb{R}^{d})^{n+1} \to \mathbb{C}$ is a continuous function. To each $\varphi \in \mathcal{S}(\mathbb{R}^{d})$, we can define the map $f[\varphi]: \mathcal{S}(\mathbb{R}^{d} …

I'm not sure this is an appropriate question for this site but I've tried math stack exchange and got no answers. Also, this problem arose in one of my research problems, so I'm stating it here. The …

It is usual to introduce Fréchet and Gâteaux derivatives in Banach spaces. In this context, the familiar Taylor expansion with remainder is also at hand, as you can see on the picture below taken from …

In the past months, I've been trying to understand the so-called Sine-Gordon transformation, so I've posted some questions here about this topic. I also did an extensive research about this subject, s …

Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space $l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} …

Following some suggestions on my previous posts, I'm trying to reformulate my question in a more specific way. This is a continuation of my original post. Since the mentioned post, I think I've learne …

Let $(X,\tau)$ be a locally convex space and $(X^{*},\tau_{s})$ be its topological dual space equipped with the strong topology. Denote by $S(X,X^{*})$ the collection of operators from $X$ to $X^{*}$ …

It is a known fact that Borel measures are uniquely determined by their Fourier transforms. This is the motivation for the following question. I came across the concept of a Borel transform of a Borel …

Let $A_{1},...,A_{n}$ be densely defined self-adjoint operators on a separable Hilbert space $\mathscr{H}$. Suppose these have a common dense domain $D\subset \mathscr{H}$ and satisfy commutation rela …

Let $f: \mathbb{R}^{3} \to \mathbb{R}$ be the solution of the following PDE: $$\Delta f -\frac{1}{2}h f = 0$$ where $h \in C_{c}^{\infty}(\mathbb{R}^{3})$ (compactly supported an smooth) and $f$ satis …

Physicists often use functional integrals and I'm trying to make sense of it in more precise terms. As you can see here, the functional derivative in Physics is defined in terms of Taylor expansions. …

This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a fu …

I've been trying to understand the so-called Sine-Gordon Transformation which occurs in both classical and quantum statistical mechanics. One of the most cited references on this topic seems to be Frö …