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3

$$\int_0^1 \underbrace{|k(s,t)-k(s',t)|}_{\leq \sup_{r \in [0,1]} |k(s,r)-k(s',r)|} \underbrace{|f(t)|}_{\leq \sup_{r \in [0,1]} |f(r)|} \, dt \leq \sup_{r \in [0,1]} |k(s,r)-k(s',r)| \cdot \|f\|_{\infty} \cdot \int_0^1 \, dt.$$


2

The nearest point projection onto a closed subset $E\subset \mathbb R^n$ is single-valued if and only if $E$ is convex*. In this case, the Lipschitz constant is equal to $1$. If $E$ is not convex, there is at least one point $x\in \mathbb R^n$ for which $\min_{y\in E}\|x-y\|$ is attained at more than one point. We could try to discuss the continuity of ...


2

If you have a bounded operator $A$, then the holomorphic functional calculus is always an option, and it is based on Cauchy's integral representation: $$ f(A) = \frac{1}{2\pi i} \oint_{C} f(\lambda)``\frac{1}{\lambda I-A}"\,d\lambda = \frac{1}{2\pi i} \oint_{C} f(\lambda)(\lambda I-A)^{-1}\,d\lambda. $$ The contour $C$ is any simple ...


1

This is just a thought which might be helpful but is a bit long to be a comment. Consider a $N\times N$ matrix $A$ whose $(1,1)$ entry is $|z|^2$ . Let $e_1=[1,0,\dots,0]^T$ denote first column of $N\times N$ Identity matrix. Then note that, $$\lambda_{min}(A) = \min_{x^Tx=1}~x^TAx\leq e_1^TAe_1=|z|^2$$


1

This is a comment to @ABC 's answer. I seem to have difficulties to understand things correctly, maybe it's simply too late. Here is a list of a: the alternating series of units, b: the partial sums (=using Cesaro-order 0), c: the partial sums using Cesaro-order 1, d: and the squares of $c$, (where it is focused that the ...


1

You can find $P$ so that $PGP^{-1}=T=\begin{pmatrix}\lambda_1 & * & *\\&\ddots & *\\&&\lambda_n\end{pmatrix}$ where the $\lambda_i$s are the eigenvalues and $*$ means "whatever". Then, $\exp(tG)=\exp(tP^{-1}TP)=P^{-1}\exp(tT)P$. $\exp(tT)=\begin{pmatrix}e^{t\lambda_1} & * & *\\&\ddots & ...


1

Jonas Meyer's comment gives an example of a self-adjoint $T$ whose spectrum is $[-1,1]$. Then we may take the absolute value function, for instance, or even any continuous function whose zeros accumulate in $(-1,1)$ to see that that the space of continuous functions on the spectrum contains functions which are not power series. Since $C(\sigma (T))$ is ...


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In the following, I will assume that $\mu$ is $\sigma$-finite for simplicity. One can probably drop that assumption with a bit of extra work. Also, I do not claim that my proof is one of the simpler ones. Let us define $$ \left\Vert k\right\Vert _{L^{p,q}}:=\left(\int\left(\int\left|k\left(x,y\right)\right|^{p}d\mu\left(x\right)\right)^{q/p}\, ...



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