# Tag Info

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• 43.4k
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### Inverse of a bounded operator on a Hilbert space

Consider $H = \ell_2(\mathbb{N})$, and let $T$ be the shift operator $T(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$. Then $\Vert Tx \Vert = \Vert x \Vert$ for all $x$, but $T$ is not surjective so it ...
• 4,188
1 vote
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• 12.5k
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### Approximation of piecwise linear function where $f_n(x)=0$

There are a few ways to resolve this, but generally your goal will be to define a sequence of functions $f_n \in C^\infty$ (or $C^3$, or whatever) such that $f_n = f$ outside $(-1/n, 1/n)$ and there ...
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### Are Fourier transforms over $L^1(\mathbb{R})$ themselves in $L^1(\mathbb{R})$?

Consider the Fourier-transformation on $L^1(\mathbb R)$ given by $$F^1 \colon L^1(\mathbb R) \to C_0(\mathbb R), \quad f \mapsto \widehat f.$$ The fact that $F^1$ maps into $C_0(\mathbb R)$ is usually ...
• 4,712
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### Inequality with norm in Space $L^2(\Omega)$

Let $A\subset \Omega$ satisfy $0<|A|<1.$ For $u= |A|^{-1/4}1\hspace{-2.5pt}{\rm I}_A$ we have $$\|u^2\|_2=1,\quad \|u\|_2=|A|^{1/4}<1$$ Hence for any constant $k>0$ the inequality does ...
• 9,111
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• 14.1k
1 vote
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• 9,111
1 vote
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### Finding the spectrum of $(Tf)(t)=\max[0,\cos(t)]f(t)$ for $f\in L^2[-\pi,\pi]$

The point spectrum $\sigma_p(T)$ consists of all $\lambda$ for which there exists $f\in L^2[-\pi,\pi]\setminus\{0\}$ such that $Tf=\lambda f$. It is not hard to see that $0$ is the only value in the ...

### Given a derivative, say, as a tensor, is the corresponding linear approximation unique?

What are the hidden assumptions in this interpretation? There are no hidden assumptions. Here is how to do this calculation: if $A$ is an invertible matrix, then for $\varepsilon$ a sufficiently ...
• 377k
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### Scaling properties of Hölder seminorms

Let $X$ be defined on $[0,1]$ by $X=X'\circ\varphi$ with $X'$ defined on $[s,t]$ and $\varphi:[0,1]\to[s,t],\;x\mapsto s+(t-s)x$. Then, $\varphi$ is increasing and onto, hence \begin{align}\|X'\|_{\...
• 3,041
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### Direct sum of Hilbert modules

You can make $E$ into a $C^\ast$-module over $A\oplus B$ by considering the action $e(a,b)=ea$ and the new inner product $\langle e_1\mid e_2\rangle=(\langle e_1,e_2\rangle,0)$. If you do the same ...
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### Nonlinear function continuous but not bounded

I have two questions on the above example for $C[0,1]$ of orangeskid (apologies, if this is misplaced, I may change this into a new question, but I thought it would be easier to read here) (1) I ...
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### Relatio between Frechet derivative and partial derivatives in $\mathbb R^n$

(We assume we are in $\mathbb{R^{n}}$). If all partial derivatives exist and are continuous, then is Frechet differentiable (since $\bigtriangledown f(x)$ exists and it is continuous) and also $C^{1}$....
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