# Tag Info

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### Proof of the completion of a normed space

Recall that elements of $\hat{X}$ are equivalence classes of Cauchy sequences in $X$, with the equivalence relation $[x] \sim [y]$ iff $x_n - y_n \to 0$, i.e $\lim_{n \to \infty}\|x_n - y_n\| = 0$. ...
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### Do we have Arzela-Ascoli theorem on the space of all continuous stochastic process?

$X_k$ need not have a subsequence that converges in $U$. One way to generate a counterexample is to consider constant processes (i.e. processes such that $t \mapsto X_{k,t}(\omega)$ is constant for ...
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### Commutant of a set of operators and norm topology.

Note the weak topology is weaker than the strong topology which in turn is weaker than the norm topology. So saying the commutant is WOT closed implies it is closed in the other two (this is in ...

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### Is the weak derivative a Radon-Nikodym derivative?

There is indeed a connection there, but your require a bit more regularity. You actually need $f\in W^{1,\infty}_{loc}(\mathbb{R})$, i.e. $f$ has to be locally Lipschitz. Then by Rademachers Theorem (...
### Fredholm Integral Equation of the Second Kind in $L_2[0,1]$
Let's first consider the simplest case: if $\lambda=0$, the solution to the integral equation $$x(t) + \lambda \int_{0}^{1}\left(\frac{1}{2} - |t-s|\right)x(s)\, ds = \cos(\pi t) \tag{1}$$ is ...