## 0.5 Geometry

The inner product brings geometry: the **length**, or **norm**, of **orthogonal** if ^{13} forms an orthonormal basis, and so any vector can be expressed as a linear combination of the basis vectors:
**dual basis**

To make the notation a bit less cumbersome, we will sometimes label the basis kets as *do not confuse |0\rangle with the zero vector*!
We

*never*write the zero vector as

Now that we have some notion of geometry, we can explain a bit more about this idea of associating a Hilbert space to a quantum system — we will use some terminology that we have not yet introduced, but all will be explained in due time.

To any *isolated* quantum system, which can be prepared in **perfectly distinguishable** states, we can associate a Hilbert space

We note here one more fact that also won’t yet make sense, but which won’t hurt to have hidden away in the back of your mind.

The inner product **probability amplitude** that a quantum system prepared in state

That is, consider sets of vectors

|e_i\rangle such that\langle e_i|e_j\rangle=\delta_{ij} (where the**Kronecker delta**\delta_{ij} is0 ifi\neq j , and1 ifi=j .), and then pick any of the largest such sets (which must exist, since we assume our vector spaces to be finite dimensional).↩︎