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Chapter 1 is mostly a review of the theory of vector spaces. As such it nothing extremely interesting or new and therefore I have little to comment here.
However the last section of the chapter is concerned with giving a kind of definition of Hilbert Spaces, which I found to be more confusing than helpful. Therefore, a few comments on this section lest I forget what (I think) I already know.
A Hilbert Space, or rather the mathematical concept of a Hilbert Space generalises the notion of an Euclidian Space, and vector algebra, from a 2 or 3 dimensional space, and vector to an infinite one. In formal terms, a Hilbert Space is an inner product space which is complete.
To put it in simpler terms, an inner product space is an abstract vector space where angles and distances can be measured. By 'complete' we mean that if a vector series in a space approaches a limit, that limit will exist in the space as well.
A few examples to clarify the concept of completeness:
The space Q of rational numbers, with the standard metric given by the absolute value, is not complete. Consider for instance the sequence defined by x1 = 1 and xn+1 = xn/2 + 1/xn. This is a complete sequence of rational numbers, but it does not converge towards any rational limit: Such a limit x of the sequence would have the property that x² = 2, but no rational numbers have that property. But considered as a sequence of real numbers R it converges towards the irrational number , the square root of two.
The open interval (0,1), again with the absolute value metric, is not complete either. The sequence (1/2, 1/3, 1/4, 1/5, ...) is complete, but does not have a limit in the space. However the closed interval [0,1] is complete; the sequence above has the limit 0 in this interval.
It is worth mentioning that intuitively a complete a complete set is one that had "no holes" in it, or that no points are missing from it, whether that happens inside the set or at the boundaries.
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