Quotient quadratic space

Let $𝑉$ be a quadratic space and $𝑈 ⊴ 𝑉$ be a normal subspace. The quotient quadratic space $𝑉 / 𝑈$ is the corresponding quotient vector space with the (well-defined) quadratic form #m/def/geoalg

𝑞 (𝑣 + 𝑈) = 𝑞 (𝑣)

Properties

The quotient vector space $𝑉 / 𝑈$ has a well-defined quadratic form iff $𝑈$ is a normal normal subspace.

Proof of 1.

For the quadratic form to be well defined, we require $𝑞 (𝑣 + 𝑢) = 𝑞 (𝑣)$ for all $𝑣 \in 𝑉$ and $𝑢 \in 𝑈$ . Equivalently

0 = 𝑞 (𝑣 + 𝑢) - 𝑞 (𝑣) = 𝑏 𝑞 (𝑣, 𝑢) + 𝑞 (𝑢)

for all $𝑣 \in 𝑉$ and $𝑢 \in 𝑈$ . This includes, however, that $0 = 𝑞 (0 + 𝑢) = 𝑞 (𝑢)$ , so any such $𝑢$ must be both degenerate and isotropic.

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