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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10
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1answer
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+100

$f(nx)\to 0$ as $n\to+\infty$

Let $f:\mathbb R^+\to\mathbb R^+$ be a continuous function that has the following property: For any $x\in I$, $f(nx)\to 0$ as $n\to+\infty$. Intuitively, if $I$ is 'big' enough, $f$ necessarily ...
4
votes
0answers
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+50

Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
0
votes
0answers
32 views
+50

Holomorphic semigroup can be extended to a strongly continous semigroup on $L^{p}$?

I have a question about $C_{0}$-semigroup theory. Let $(H,(\, , \,))$ be a real Hilber space and $(H_{\mathbb{C}}, (\, , \,))$ the its complexification. Any linear operator $(L,D(L))$ on $H$ can be ...
2
votes
1answer
45 views
+50

Representation of the Fréchet derivative of $〈f,e_n〉$, where $f:H→H$, $H$ is a Hilbert space and $(e_n)_{n∈ℕ}$ is an orthonormal basis of $H$

Let $H$ be a $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $f:H\to H$ be Fréchet differentiable and $$f_n:=\langle f,e_n\rangle\;\;\;\text{for }n\in\mathbb N$$ ...