How generalize the definition of normed linear space.

Hello.
I want to know how the definition of normed linear space generalized.

As we know normed linear space(X) is a linear space with ||*||:X->R
norm, which satisfy
1) ||x|| >= 0
2) ||cx|| = |c| ||x||
3) ||x+y|| <= ||x|| + ||y||
4) ||x||=0 <=> x=0.

we can generalize it.

X group over C filed is called linear space if for a,b,1 in C and x,y
in X
1)a(bx))=(ab)x
2)1*x = x
3)(a+b)x = ax +bx
4)a(x+y) = ax+ay

X is called normed linear space if it is linear space and ||*||:X->R,
where R is set, and there is relation (linear ordering) on R - '>='.
1) ||x|| >= 0
2) ||cx|| = |c| ||x||
3) ||x+y|| <= ||x|| + ||y||
4) ||x||=0 <=> x=0.

But there are some questions about
1) how we define |c| for c in C.(what condition this action must
satisfy?)
2) how we define * action between c in C and r in R.(what condition
this action must satisfy?).


Thank You
MKo

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