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A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.
15
votes
Accepted
Can an operator have Exp(z) as its characteristic "polynomial"?
Here's a proof that $\exp(z)$ is not a characteristic function using the product expansion for the determinant, which is essentially equivalent to Lidskii's theorem stating that the trace of a trace c …
23
votes
Gaussian processes, sample paths and associated Hilbert space.
No, it is not true for simple examples such as standard Brownian motion or a sequence of independent random variables.
Suppose $W$ is a standard Brownian motion on the interval $[0,T]$. The covarian …
10
votes
Does there exist an event independent of a given sigma-algebra?
No. If the probability space contains atoms then you can easily construct sub-$\sigma$-algebras which are not independent to any nontrivial events.
However, we can still construct examples, even for …