A note on the Hilbert Space

This blog post ties off loose ends.
As has been laid out, there is a direct correspondence between the usual picture of wave functions,Schrödinger’s equations and kets and bras introduced by Paul Dirac.
We have already seen that we interpret what happens in Quantum Mechanics as simply being a collection of Hermitian Operators acting on some vectors. For any experiment what we measure are the eigenvalues and the statistics we collect of experiments are simply the expectation values of these operators. So a question we can ask is where do these vectors "live". From physical intuition we know we need some notion of "dot product" , (we are incidentally using our intuition from the normal Euclidean space). For this dot product the order of operators in the dot product should not matter i.e \( \vec{v}.\vec{w} = \vec{w}.\vec{v} \). For the moment we shall not worry about notation. Also we require that \( \vec{v}.\vec{v} \) be positive definite and only zero when \( \vec{v}=0 \). Lastly, we may require linearity namely \( \vec{u}.( \vec{v} + \vec{w})  = \vec{u}.\vec{v} + \vec{u}.\vec{w}\).
In all this we are assuming all the usual properties that are assumed for distance functions and norms. The vectors  obviously constitute a vector space.  Thus what we really have is an inner product space. Lastly, we also have a metric space with additional property that all Cauchy sequences converge in this vector space.

Note: A metric space is a topological space with a distance function defined upon it. At some point I shall have more to say about this.

Thus we can now say what we mean by a Hilbert space by saying it is an inner product space that is also a complete metric space. Not that this definition does not care what our actual vectors look like. In fact if one goes through the desired properties of our dot product one can easily see that these are satisfied by wave functions with the dot product provided by the integral or by what we usually mean by vectors namely column and row things. In fact the usual n-dimensional real space is also a Hilbert space. In other words there are different kinds of Hilbert spaces; the kind we assume when we write down wave functions obeying a differential equation live in a specific kind of Hilbert space called \(  L^2 \) space. But there is another incarnation of Hilbert space we can use that is provided by bras and kets, as far as I know it has no special name.
These two kinds of Hilbert spaces can roughly be thought of as choosing what sort of basis you want. If you choose a continuous basis you arrive at an \( L^2\) space and generally speaking when we use bras and kets we are most likely working in a discrete basis. This connection shall be explored in greater detail once Representation theory is discussed.

Notation: Dirac introduced  \( | v \rangle \) and  \( \langle w| \) to represent the vectors in our Hilbert spaces. For finite dimensional Hilbert space one can think of these as row and column vectors although technically what we have is an element from a vector space,  \( |v \rangle\) and  \( \langle w| \) an element from the dual vector space (a space of linear functionals). Both these vectors spaces in our case have the same dimensions although in general they need not be.