At the beginning I was only annoyed by all the question marks that were placed instead of the citations. This happens because the default setting for the preview is one compilation. I could easily fix that by going to Preferences and changing the number of typesetting runs to 2. This did the job.

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I have always thought that double strike letters like , , were some smart invention to give sets we find so incredibly important a clever distinguished notion. It turns out the evolution of this notation was quite a coincidence. These letters originated from the blackboard – formerly the sets of natural, real, rational numbers and so on were denoted by bold letters (which you can actually still find in old textbooks). However, writing bold letters on the blackboard – well that’s beyond our artistic skills (or just very inconvenient) and therefore people started to write double strike letters on the blackboard to represent bold letters. And since mathematicians are always short of symbols, people adapted these letters to printed texts too. Now this explains the command \mathbb in – math blackboard bold. Some people think it is not ok to use this notation, there is even an entire Wikipedia article about this subject. ]]>

Here is a sufficient condition for a linear operator to be bounded:

Theorem (Hellinger-Toeplitz theorem)

Let be an everywhere defined linear operator on a Hilbert space with for all and in . Then is bounded.

This theorem states that an everywhere defined linear operator on a Hilbert space that is symmetric everywhere on is always bounded. The Hellinger-Toeplitz theorem implies that an unbounded symmetric operator cannot be defined on all of ! It tells you that when dealing with unbounded operators it is very important to specify the domain on which the operator is defined.

The Hellinger-Toeplitz theorem is an immediate consequence of the closed graph theorem. The graph of a linear operator is defined as follows:

Definition (Graph)

Let be a linear map from one normed linear space into another. Its graph is the set

The closed graph theorem states:

Theorem (Closed graph theorem)

Let be a linear operator mapping from one Banach space into another. Then is bounded if and only if its graph is closed.

One can show that for an everywhere defined linear operator the symmetry implies that the graph is closed and hence, the closed graph theorem gives that the operator is bounded.

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