References: edX online course MIT 8.05.1x Week 4.
Sheldon Axler (2015), Linear Algebra Done Right, 3rd edition, Springer. Chapter 6.
An inner product defined on a vector space is a function that maps each ordered pair of vectors of to a number denoted by . The inner product satisfies the following axioms:
- positivity: for all .
- definiteness: if and only if (the zero vector).
- additivity in the second slot: .
- homogeneity in the second slot: where .
- conjugate symmetry: .
[Axler requires additivity and homegeneity in the first slot rather than the second, but Zwiebach uses the conditions above, which are more usual for physics.]
The norm of a vector is defined as
All the above applies to both real and complex vector spaces, although it should be noted that in the case of conjugate symmetry in a real space, so in that case, the condition reduces to .
Conditions 3 and 4 apply to the first slot as well, though with a slight difference for complex vector spaces. [These properties can actually be derived from the 5 axioms above, so aren’t listed as separate axioms]:
Two vectors are orthogonal if . By this definition, the zero vector is orthogonal to all vectors, including itself.
The inner product is non-degenerate, meaning that any vector that is orthogonal to all vectors must be zero.
An orthogonal decomposition is defined for a vector as follows: Suppose and . We can write as
From the definition, so we’ve decomposed into a component ‘parallel’ to and another component orthogonal to .
There are a couple of important theorems for which we’ll run through the proofs:
Theorem 1 Pythagorean theorem. If and are orthogonal then
Proof: From the definition of the norm
where in the third line we used the orthogonality condition .
Theorem 2 Schwarz (or Cauchy-Schwarz) inequality. For all vectors
Proof: The inequality is obviously true (as an equality) if , so we need to prove it for . In that case we can form an orthogonal decomposition of :
Since and are orthogonal, we can apply the Pythagorean theorem:
Rearranging the last line and taking the positive square root (since norms are always non-negative) we have
Theorem 3 Triangle inequality. For all
Proof: As with the Schwarz inequality, we start with the square of the norm:
The last line follows because is a complex number, so
We can now apply the Schwarz inequality to the last term in the last line to get
Taking the positive square root, we get
A finite-dimensional complex vector space with an inner product is a Hilbert space. An infinite-dimensional complex vector space with an inner product is also a Hilbert space if a completeness property holds. This property is a technical property which is always satisfied in quantum mechanics, so we can assume that any infinite-dimensional complex vector spaces we encounter in quantum theory are Hilbert spaces.