Dual spaces are coming up in my self studies now. after many sources and lecture videos I still don't feel confident I understand.
Can someone help correct any misunderstanding and or confirm my current understanding.
A dual space is a space of all functions that satisfy the definition of a vector space ( you can recycle the definition of a vector space )
Does it just serve as a definition for functions that provide basis transformations?
Functions take a vector as argument and return a new vector? or a scalar? Both possibilities? (several lectures talked about mapping vectors the real numbers. others implied it returns a full vector.
If it contains all possible vectors, but does not index them in any useful way then how is it useful. I can't use them by referencing an index number. I still have to define the function myself. so isn't "dual space" just a classification of function such that "dual space function" satisfy certain criteria.
As you can probably tell I'm a programmer. I am falling in love with math but sometimes the language and abstract nature causes seg faults in my brain lol.
Thanks in advance for any help.