# Tagged Questions

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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### Product topology on a product space of normed spaces is normable iff the product is finite [duplicate]

Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that ...
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### Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
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### How to define $E(X)$ when X is a random variable from the sample space to an infinite-dimensional topological vector space?

Let $V$ be an infinite-dimensional vector space with a topology. For simplicity, we can assume $V$ is a Banach space. Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X:\Omega \to V$ be a ...
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### Convolution of a distribution with a $C^{N}$ function.

I've been working on the following Problem from Friedlander's introduction to the theory of distributions: Show that if $u$ is a distribution of order $N$ and with compact support on ...
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### Closed, convex and balanced subset of a vector space

Let $(X,\|\ \|)$ be a vector space over $K$ and let $B\subseteq X$ be closed, convex and balanced. I want to prove the following: If $x_0\in X\setminus B\Rightarrow\exists\;f\in X^*$ s.t. ...
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### semifinite von Neumann algebra, spectral projection, trace

Let $\mathcal{M}$ be a semifinite von Neumann algebra and $\tau$ be a semifinite faithful normal trace on it. Let $T,P_1,P_2\in \mathcal{M}$, where $P_1,P_2$ are projections with $P_1\perp P_2$. Then, ...
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Doesn't the the Mercer's theorem say something stronger than just the spectral theory of compact self-adjoint operators on a Hilbert space applied to the reproducing "kernel" function? As in if I ...
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### Does a polynomial function on spectrum uniquely define polynomial on operator?

Let $X\subset\mathbb C$ be a compact set, let $T$ be a bounded operator with its spectrum contained in $X$, let $P$ be a polynomial. Is it true that whenever $P=0$ on $X$ then $P(T)=0$?
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### Find the adjoint of $-\frac{1}{2\pi i}\int_C R_z \ dz$

Let $L$ be a self adjoint linear operator (not necessarily bounded) and $C$ apositively oriented simple closed curve in the resolvent set encircling $\sigma_0 \subset \sigma(L)\subset \mathbb{R}$. ...
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### $2$-capacity of a set in $\mathbb{R}^n$

Let $B_r \subset \mathbb{R}^n$ denote the ball of radius $r$ centered at the origin. Consider any set $F \subset B_1$. For all sets $\Omega \subset \mathbb{R}^n$ such that $F \subset \Omega$, define ...
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### Finding $\|f\|$
Given $f: \ell^2 \rightarrow \mathbb R$ defined by $$f(a_1,a_2,...)=\sum_{n=1}^{\infty} \frac{(-1)^n}{3^{n-1}} a_n$$ Is $f$ a bounded linear functional? If yes, then find $\|f\|$. It is definitely ...
### Integrating $\int _a^b |f(x)| dx$
Is there a way to calculate these without having to sketch out the function first? Seems like you just plus everything when you evaluate the limits. It doesn't seem like it is simply $|g(b)|-|g(a)|$ ...