## 0.3 Bras and kets

An **inner product** on a vector space

\langle u|v\rangle=\langle v|u\rangle^\star \langle v|v\rangle\geqslant 0 for all|v\rangle \langle v|v\rangle= 0 if and only if|v\rangle=0

where

The inner product must also be *linear* in the second argument but *antilinear* in the first argument:

To any physical system we associate^{10} a complex vector space with an inner product, known as a **Hilbert space**

If **norm** on **metric** by defining **Cauchy** if its elements “eventually always get closer”, i.e. if for all **complete** if *every Cauchy sequence converges in that space*, i.e. if the limits of sequences that *should* exist actually *do* exist.

Now one useful fact is the following: on a *finite dimensional* vector space, all norms are equivalent.
(Note that this does *not* mean that *finite dimensional* vector space, the unit ball is compact.
By a short topological argument, we can use these facts to show that what we claimed, namely that every *finite dimensional* inner product space is complete (with respect to the norm induced by the inner product, and thus with respect to *any* norm, since all norms are equivalent).

In the infinite dimensional case these facts are *not* true, and we have a special name for those inner product spaces which *are* complete: **Hilbert spaces**.
So working in the finite dimensional case means that “we do not have to worry about analysis”, in that the completeness property comes for free the moment we have an inner product, and we can freely refer to inner product spaces as Hilbert spaces.

For example, for column vectors **bra** vector, or a **bra**, and can be represented by a row vector:

Bras are vectors: you can add them, and multiply them by scalars (which, here, are complex numbers), but they are vectors in the space **dual** to **linear functionals**, that is, linear maps from

All Hilbert spaces of the same (finite) dimension are isomorphic, so the differences between quantum systems cannot be really understood without additional structure. This structure is provided by a specific algebra of operators acting on

The question of

*how*exactly we construct this associated space is a subtle one in the case of arbitrary physical systems, but we shall see that this is relatively straightforward when working with the types of systems that we consider in this book.↩︎