Hilbert Spaces

The mathematical concept of a Hilbert Space is named after the famous mathematician David Hilbert and is a generalisation of an Euclidean space. It is an abstract vector space possessing the structure of an inner product allowing length and angle to be measured.

The most important space in linear algebra is the Euclidean space in n dimensions. A typical element in this space is an n-element vector of real numbers. With these vectors we can add them, find the angle between them by taking a dot product, and take limits of a sequence of vectors.

Consider taking vectors to have an infinite list of coordinates – (a1,a2,a3,…). We can add these vectors and, if we want to find the angle between two such vectors, we can try to compute the dot product:

Screen Shot 2017-06-23 at 5.40.32 PM.png

However, this dot product may not converge. Thus, to ensure it converges we must ensure that both sequences (ai) and (bi) are square-summable, i.e.

Screen Shot 2017-06-23 at 5.41.58 PM.png

Under this assumption, the dot product is in fact finite and hence we can find the angle between the two vectors using the formula:

Screen Shot 2017-06-23 at 5.45.13 PM.png

The length of a vector is defined as

Screen Shot 2017-06-23 at 5.45.24 PM.png

Now that the notion length and angle have been defined, let us talk about sequences. We say that a sequence of infinite-dimensional vectors converges to some particular vector if the lengths of their differences from that vector tend to zero. Just as for the real numbers and Euclidean space, any Cauchy sequence in this space converges, i.e. the space is complete.

The space of infinite-dimensional vectors described above define the Hilbert space Screen Shot 2017-06-23 at 5.50.27 PM.png, which is one of the simplest Hilbert spaces. Now, if instead of merely allowing your vectors to be indexed by integers one allows them to be indexed by an arbitrary set then you have obtained the most general type of Hilbert space.

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