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Shouldn't the induced norm be sqrt(e0.inner_product(e0))? I would just write a function to do this:

def mynorm(v):
    return sqrt(v.inner_product(v))

I've heard that Sage could do more to support working with non-standard inner product matrices. This might be an example of a place that could be improved.

Alternatively, if you wanted to modify the code in devel/sage/sage/modules/free_module_element.py, you could submit a patch. In the norm function, I would check to see if the inner product matrix defines the dot product:

if not self.parent().__inner_product_is_dot_product()
    special-case code for inner product matrices

Also, you might change the __abs__ function.

Shouldn't the induced norm be sqrt(e0.inner_product(e0))? I would just write a function to do this:

def mynorm(v):
    return sqrt(v.inner_product(v))

I've heard that Sage could do more to support working with non-standard inner product matrices. This might be an example of a place that could be improved.

Alternatively, if you wanted to modify the code in devel/sage/sage/modules/free_module_element.py, you could submit a patch. In the norm function, I would check to see if the inner product matrix defines the dot product:

if not self.parent().__inner_product_is_dot_product()
self.parent().__inner_product_is_dot_product():
    special-case code for inner product matrices

Also, you might change the __abs__ function.