In the same way that we defined a topological group to be a space with points that act like group elements, we can define a **topological vector space** to be a Hausdorff space with points that act like vectors over some field, with the vector space operations continuous. However, a better definition might be a vector space with a topology that makes it Hausdorff. This is because the vector space structure already contains topological information in its scalars, which interact as Euclidean spaces. This is reflected in the fact that a finite-dimensional vector space can only be made into a topological vector space with the Euclidean topology; however, this is not true for infinite-dimensional vector spaces.

Taking another approach, we can use norm and inner product structures on a real vector space to turn it into a hybrid object. Recall that a metric space is a topological space with a distance function. A **complete metric space** is one in which the limit of every Cauchy sequence (a sequence of points that become arbitrarily close) is also in the space. In a normed vector space, the norm defines a distance function \({\left\Vert v-w\right\Vert }\), which turns the space into a hybrid object, a metric/vector space. A complete normed vector space is called a **Banach space**, and an inner product space that is complete with respect to the norm defined by the inner product is called a **Hilbert space**. All finite-dimensional inner product spaces are automatically Hilbert spaces, but applications in physics typically involve infinite-dimensional spaces, where more care is required.

Introducing the multiplication of vectors, a **Banach algebra** is a Banach space that is an associative algebra satisfying \({\left\Vert vw\right\Vert \leq\left\Vert v\right\Vert \left\Vert w\right\Vert }\). The concept of a conjugate is generalized in a ***-algebra**, a complex associative algebra with an anti-linear mapping * that is both an **involution**, i.e. applied twice to any element it is the identity \({v^{**}=v}\), and an **anti-automorphism**, i.e. it is an automorphism except under multiplication where we require that \({(vw)^{*}=w^{*}v^{*}}\). A *-algebra that is also a Banach algebra is called a **B*-algebra**. Finally, if the property \({\left\Vert vv^{*}\right\Vert =\left\Vert v\right\Vert ^{2}}\) also holds for all vectors, it is called a **C*-algebra**.

The main use of all this in physics is in quantum theories. Hilbert spaces are restrictive enough to act the most like finite-dimensional vector spaces, and the algebra of continuous linear operators on a complex Hilbert space is a C*-algebra. This line of reasoning leads us into analysis, a part of mathematics we will not address in this book; however, here we list some relevant facts for a Hilbert space \({H}\), with details omitted:

- Every Hilbert space admits an orthonormal basis (where every element of \({H}\) is a possibly infinite linear combination of basis vectors)
- Most Hilbert spaces in physics are
**separable**, meaning they have a countable dense subset - All separable Hilbert spaces of countably infinite dimension are isomorphic; thus the references in physics to “Hilbert space”
- Any closed subspace of a Hilbert space has an orthogonal complement
- The dual space \({H^{*}}\) (the space of all continuous linear functions from \({H}\) into the scalars) is isomorphic to \({H}\)
- Every element of \({H^{*}}\) can be written \({\left\langle v,\;\right\rangle}\) for some \({v\in H}\) (sometimes called the
**Riesz representation theorem**) - This justifies the
**bra-ket notation**(AKA Dirac notation), in which we write \({\left|\psi\right\rangle \in H}\), \({\left\langle \psi\right|\in H^{*}}\), so that in terms of a basis \({\left|\varphi\right\rangle }\) we have \({\left|\psi\right\rangle =\sum\left|\varphi_{i}\right\rangle \left\langle \varphi_{i}\left|\psi\right.\right\rangle }\)