Franklin's Notes


Metric vector space

A metric vector space is a vector space $X$ over $\mathbb R$ with a metric tensor, that is, a symmetric nondegenerate bilinear form $G:X\times X\to \mathbb R$. The quantity $G(x,y)$ for two vectors $x,y\in X$ is often abbreviated $x\cdot y$. If $G$ is positive or negative definite, then this is called an inner product vector space.

In a metric vector space, the length of a vector, denoted $|x|_G$, is defined as the quantity $\sqrt{x\cdot x}$. Notice that this quantity may be imaginary, as $x\cdot x$ may be negative if $G$ is indefinite or negative definite. To borrow language from physicists, we say that $x$ is

Inner products on a vector space induce a norm, but metric tensors only induce a partial norm (i.e. a function satisfying all norm axioms except for $\lVert x\rVert = 0\implies x=0$, which is relaxed to $\lVert x\rVert \ge 0$). The partial norm $\lVert \cdot \rVert_G$ induced by a metric tensor is defined by

geometry

vector-space

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