A * Hilbert space* is a complete inner product space. It allows the application of linear algebra and calculus techniques in a space that may have an infinite dimension.

The inner product in a Hilbert space has the following properties:

- Conjugate symmetry:
- Linearity with respect to the 2
^{nd}argument: - Antilinearity with respect to the first argument:
- Positive semi-definiteness: , with if

The last property can be illustrated using the space that is equipped with an inner product. Such a space is an example of a real finite-dimensional Hilbert space. The inner product of the vector with itself is:

We define a complete Hilbert space as one where ** every Cauchy sequence** in converges to an element of . If you recall, a Cauchy sequence is a sequence, e.g. where , for which

We can also define the completeness of a Hilbert space in terms of a sequence of vectors , where . Each element is represented by a series of vectors, which converges absolutely (i.e. ) and converges to an element of . In other words, the series of vectors in converges to some limit vector in :

Generally, every element of a vector space can be a point, a vector or a function. In quantum mechanics, we are interested in a Hilbert space called the space, where the eigenfunctions of a Hermitian operator are * square integrable*, i.e. .

Not to be confused with the completeness of a Hilbert space, the ** completeness of a set of basis eigenfunctions** refers to the property that any eigenfunction of the Hilbert space can be expressed as a linear combination of the basis eigenfunctions. An example is the space, where the set of basis vectors is complete, with a linear combination of and spanning . In , the number of basis vectors

**may be infinite. If the set of**

**is complete, we say that it spans , which is itself complete.**

Just as the orthonormal vectors and form a complete set of basis vectors in the space, where any vector in can be expressed as a linear combination of and , we postulate the existence of a complete basis set of orthonormal wavefunctions of any Hermitian operator in .