# 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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### Prove that the set $X+Y$ is closed

Consider two sets: $$X=\bigl\{(x,y)\in \mathbb R^{2}:xy=1\bigr\}$$ $$Y=\bigl\{(x,y)\in \mathbb R^{2}:|x|\le 1,|y|\le 1\bigr\}.$$ We find that , both the sets are closed. But, is the set $X+Y$ ...
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### Size of function spaces

For example, how big is the space $C^k(\mathbb{R},\mathbb{R})$ ? How much is, say, $C^0$ larger than $C^1$ ? How can one figure out ?
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### Laplace Transform of $e^{a t^2}$

What is the Laplace transform of $e^{a t^2}$, for positive $a$? In order for Laplace transform to exist function must be locally integrable. Since integral of any compact set $e^{a t^2}$ is finite ...
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### Is Fourier transform method suitable for solving equation $\int g(x-t)e^{-t^2} dt = e^{-a|x|}$

Is Fourier transform method suitable for to solve the following equation \begin{align*} \int g(x-t)e^{-t^2/2} dt = e^{-a|x|} \end{align*} Suppose we take the Fourier transform of the above ...
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### Spectral definition of (fractional) Laplacian, need help understanding text

Let $\varphi_k$ and $\lambda_k$ be the eigenfunctions and eigenvalues of the Dirichlet Laplacian $-\Delta$ on some bounded domain $\Omega$. We know $\varphi_k$ are smooth and form an orthogonal basis ...
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### Compact operators and essential spectral radius

Let $E$ be an infinite-dimensional complex Banach space. Let $\mathcal{L} (E)$ be the space of endomorphisms of $E$, endowed with the operator norm. Then $\mathcal{L} (E)$ is a unital Banach algebra ...
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### General “Hodge theorem”

I know basically zero Hodge theory, so this question might be weird. Let $$A \stackrel{S}{\longrightarrow} B \stackrel{T}{\longrightarrow} C$$ be a sequence of closed, densely defined maps of ...
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### Is the Riemannian distance function Lipschitz on a hypersurface?

Let $M$ be a compact hypersurface in $\mathbb{R}^{n+1}$ of dimenion $n$. Is it true that there exists a constant $C$ such that $$d(x,y) \leq C|x-y|$$ for all $x, y \in M$? Here $d$ is the Riemannian ...
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### Equivalent finite subspaces of a hilbert space

I have to prove the following statement: Let $H$ be a Hilbertspace and $M,N$ closed subspaces. Then the following holds: If $M \sim N$ and $N$ is finite, then $M$ is finite. I think it should say ...
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### Why is $R-\lambda$ invertible for $|\lambda|<1$

I got the following question: Why is $R-\lambda$ invertible for $|\lambda|>1$ and not invertible for $|\lambda|\leq1$ ? R is the right shift operator on $\mathfrak{l^2}$
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### Prove subset of $\mathbb{C}^n$ is convex and complete

I have to prove that the subset $M=\sum_{i=1}^n x_i=1$ of $\mathbb{C}^n$ is convex and complete w.r.t. the inner product $<x,y>=\sum_{i=1}^n x_i\bar{y_i}$. Now being convex is trivial. However ...
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### weak* convergence definition in Sobolev space

I have a question which might quite trivial but I would appreciate any assistance. Why does it follow that for Sobolev spaces, say $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$, it follows ...
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### Dual space of L infty space

The Dual space of L infty space is not L1 ,are there some example to show this? I am going to use rieze representation thm,but it can not be used because p= $\infty$.
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### Mollifier proof, Evans PDE [duplicate]

My question is related to the mollifier properties in the appendix of Evans' PDE. In proving that $f^{\varepsilon}$ is smooth, he constructs the difference quotient, with the original integral over \$...