I own an M4 iPad Pro and can't figure out what to do with even a fraction of the horsepower, given iPadOS's limitations. The rumors about an upcoming touchscreen Mac are interesting, perhaps Apple will deign to make their ridiculously overpowered SOCs usable for general purpose computing. A man can dream..
> I own an M4 iPad Pro and can't figure out what to do with even a fraction of the horsepower, given iPadOS's limitations.
Literally everything you do gets the full power of the chips. They finish tasks faster using less power than previous chips. They can then use smaller batteries and thinner devices. A higher ceiling on performance is only one aspect of an upgraded CPU. A lower floor on energy consumed per task is typically much more important for mobile devices.
It is just unhelpful in many ways. It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces.
In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. They are not defined that way.
The right didactic way, in my experience, is introducing vector spaces first. Vectors are elements of vector spaces, not because they can be written in any particular basis, but because they fulfill the formal definition. And because they fullfil the formal definition they can be written in a basis.
Except just about all relevant applications that exist in computer science and physics where fixating on a representation is the standard.