Overview
An continuous dimensional hilbert space is a space where the basis of the space is isomorphic to the real numbers. That is, there is a function
{% \phi : \mathbb{R} \rightarrow E %}
where {% E %} is a set of basis elements of the space.
Using Dirac Notation, a basis element can be written as {% | x \rangle %}. That is,
{% \phi(x) = |x \rangle %}
Then, an element of the Hilbert space can be written as an
integral.
{% |f \rangle = \int_{- \infty}^{\infty} dx f(x) |x \rangle %}
Inner Product
The inner product of two basis elements, is then the Dirac Delta Function.
{% \langle x' | x'' \rangle = \delta(x' - x'') %}
{% \langle x' | f \rangle = \int_{- \infty}^{\infty} dx f(x)\langle x' | x \rangle =
\int_{- \infty}^{\infty} dx f(x) \delta(x - x') = f(x')
%}