What remains to be verified to prove that the averaged form is a valid inner product?

Conditions

• The averaged form is defined as $(v, w) := \frac{1}{|G|} \sum_{g \in G} \langle \rho(g)v, \rho(g)w \rangle$.
• $\langle \cdot, \cdot \rangle$ is a valid inner product on $V$.
• $G$ is finite.

Reasoning, step by step

1. Acknowledge that the construction preserves linearity in each argument due to the linearity of $\rho(g)$ and the initial inner product.
2. Identify that symmetry/conjugate symmetry is also preserved by the averaging process.
3. Recognize that positive definiteness is not automatically obvious from the sum alone without careful argument.
4. Consult the video's note which assigns "(Exercise: show (v,w) inner product)" specifically addressing the positivity axiom.
5. Conclude that proving $(v, v) > 0$ for $v \neq 0$ is the remaining task.

Example

The video states that bilinearity is obvious but positivity must be proved, assigning it as an exercise: "(Exercise: show (v,w) inner product)".

Common misconceptions

• Assuming that because the form is built from inner products, it is automatically an inner product without checking positivity.
• Thinking that the averaging process might destroy positive definiteness; it actually preserves it, but the proof is required.

Watch the explanation

Explore next

• How does the positive definiteness of the averaged form rely on the properties of the initial inner product?
• Why is the finiteness of $G$ important for the definition of the average?