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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 …
George Lowther's user avatar
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 …
George Lowther's user avatar
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 …
George Lowther's user avatar