# Tagged Questions

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### Compactness of the Volterra opelator

The Volterra operator is given as \begin{eqnarray} (Vf)(x)=\int_0^xK(x,y)f(y)\,{\rm d}y. \end{eqnarray} By the ArzelĂ â€“Ascoli theorem, $V\colon C^0[0,1]\rightarrow C^0[0,1]$ is compact operator. But, ...
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### Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
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### Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
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### Show compactness/ noncompactness of an operator by approximation

I have to show whether the following operator is compact or not: $$T\colon\ell^2\to\ell^2: (x_n)_{n\in\mathbb{N}}\mapsto\left(\frac{x_n+x_{n+1}}{2}\right)$$ My idea was to approximate $T$ by ...
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### Determine operator norm and show compactness

Consider $$T\colon\ell^1\to\ell^1, (s_n)\mapsto\left(\frac{s_{n+1}}{n}\right).$$ Calculate the norm of $T$ and show that $T$ is compact. 1.) Operator norm of $T$ What I have is the ...
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### Show non-compactness of multiplication operator on $C[0,1]$

Show that the multiplication operator $$(Ax)(t):=(t+1)x(t)$$ in the Banachspace $C[0,1]$ is not compact. Again I am struggling with compactness, it is always difficult to me to decide ...
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Hell0 there! I have to show whether the operator $$T\colon L^2(\mathbb{R})\to L^2(\mathbb{R}), f\mapsto\chi_{[0,1]}f$$ is continuous, selfadjoint and compact. I have problems to show the ...
Consider the heat equation $$u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$ with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$ and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$. 1.) ...
In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...