# 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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### Fabian. Functional analysis and infinite dimensional geometry

Where can I find solutions to the problems of Fabian. Functional analysis and infinite dimensional geometry, 3 - Weak Topologies
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### A canonical example for spaces that aren't $1^{\text{st}}$ countable

To get the feeling for metric spaces (or $1^{\text{st}}$ countable spaces, in general), I always find that visualizing them as $\Bbb R^3$ gives a good intuition of what is going on (locally, of course)...
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### Line in Modified Bessel Function of the First Kind

Plotting the modified Bessel function of the first kind $I_\nu(x)$ as a function of two real variables, it looks like to one side of $x=\frac{2}{3}\nu$ the function falls rapidly to zero and on the ...
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### Eigenvalue of every eigenvector is an eigenvalue of element of o.n eigenvector basis

I need to prove (or disprove) that given a bounded operator $A$ on a Hilbert space with orthonormal basis of eigenvectors $\left\{ e_i \right\}$, if $Av=\lambda v$ for $v\neq 0$ then $\lambda$ is an ...
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### Is diameter of a set a measure?

Suppose the diameter of a nonempty set $A$ is defined as $$\sigma(A) := \sup_{x,y \in A} d(x,y)$$ where $d(x,y)$ is a metric. Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable ...
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Let $f\in L^1(\mathbb{R})$ where the measure is taken to be the Lebesgue measure. The Fourier transform of $f$ is the function $\hat{f}$ defined as $$\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{... 0answers 28 views ### The L^2 convergence of semi-p-lapace equation This question is similar to the one I post early here. But this one might be more reasonable I think... Let g\in L^\infty(\Omega) be given, where \Omega\subset \mathbb R^2 is open bounded with ... 0answers 28 views ### The convergence of p-laplace equation Let g\in L^\infty(\Omega) be given, where \Omega\subset \mathbb R^2 is open bounded with smooth boundary. Define, for 1<p\leq 2,$$ u_p:=\operatorname{argmin}\left\{\int_\Omega|u-g|^pdx+\...
Let $\mathcal{H}$ be a Hilbert space, and $A$ a self-adjoint operator with domain $D_{A} \subseteq \mathcal{H}$. Assume that $\lambda_0 \in \rho(A)$, where $\rho(A)$ is the resolvent set of $A$. For ...