Insights Hilbert Spaces And Their Relatives - Operators

[fresh_42]

Operators. The Maze Of Definitions.​

We will use the conventions of part I (Basics), which are $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, $z \mapsto \overline{z}$ for the complex conjugate, $\tau$ for transposing matrices or vectors, which we interpret as written in a column if given a basis, and $\dagger$ for the combination of conjugation and transposition, the adjoint matrices. $\mathcal{H},\mathcal{H}_1,\mathcal{H}_2,\ldots$ indicate Hilbert spaces. Their dual spaces are noted by $\mathcal{H}^*$, the orthogonal complements of a subspace $U$ as $U^{\perp}$. Our inner products will be sesquilinear in the first and linear in the second argument. Integrability usually refers to the Lebesgue measure.

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[jedishrfu]

Great article @fresh_42 !

 

[fresh_42]

Great article @fresh_42 !

Thanks, but I find it a bit boring. So many different aspects only to describe a linear function. I hope that at least the list at the end is of some help to look up definitions in a short time. I hope the next part will be a bit more exciting, i.e. more examples than theory. However, one needs the vocabulary first.

 

[Mark44]

Nice work, fresh! (Notwithstanding that you find it boring!)