## 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 .

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

With any physical system we associate a complex vector space with an inner product, known as a **Hilbert space**^{3}

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 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 term “Hilbert space” used to be reserved for an infinite-dimensional inner product space that is

**complete**, i.e. such that every Cauchy sequence in the space converges to an element in the space. Nowadays, as in these notes, the term includes finite-dimensional spaces, which automatically satisfy the condition of completeness.↩︎