"Length" function without absolute homogeneity property? A norm $\|\cdot\|$ must have the property of absolute homogeneity. I'm working with a function that acts like a "length," but which can also include negative numbers (so "length" is used loosely here, but it still makes sense in my context). I'd like to use the $\|\cdot\|$ notation as it's quite simply notationally convenient and easy to read, but really I don't feel justified as my function does not satisfy $f(ax)=|a|f(x)$ in general. Is there a generalisation of norm which excludes absolute homogeneity? Or must I just define a function $f$ with the properties I want and not use the $\|\cdot\|$ notation?

If your "length" function is actually a quadratic form, you can use the notation:
$$q(x) = a_{11}x_1^2 + a_{12}x_1x_2 + \cdots + a_{nn}x_n^2$$

If your "norm" however is subadditive and separates points, but the only thing it doesn't have is the "absolute" part of absolute homogeneity, then I don't think there's a word for that concept (but don't quote me on that).  Both the semi- and quasinorms are defined differently.
