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

where

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 1Pythagorean theorem. If and are orthogonal then

*Proof:* From the definition of the norm

where in the third line we used the orthogonality condition .

Theorem 2Schwarz (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 3Triangle 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.

Mark WeitzmanActually I think the infinite dimensional spaces in QM frequently fail to be Hilbert Spaces (hence Dirac delta function etc.). But they are usually treated under an extension called rigged Hilbert Spaces. See Quantum Mechanics a Modern development by Ballentine. But essentially you are correct in that we can use the Hilbert space formalism in infinite dimensional QM spaces, if we are willing to overlook certain technicalities – like non-normalizable states etc.

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