# Normed linear Space

• Jun 1st 2010, 12:40 AM
neset44
Normed linear Space
I really need help for this . don't know where i need to start

Question is
Prove that C[a,b] is a normed linear space with ||f||= sup|f(x)| x is an element of closed interval [a,b] .

#Here C[a,b] denotes the set of all continuous real valued functions defined on the closed interval [a,b].
• Jun 1st 2010, 01:27 AM
HallsofIvy
Quote:

Originally Posted by neset44
I really need help for this . don't know where i need to start

Question is
Prove that C[a,b] is a normed linear space with ||f||= sup|f(x)| x is an element of closed interval [a,b] .

#Here C[a,b] denotes the set of all continuous real valued functions defined on the closed interval [a,b].

The problem, then, is just to show that all parts of the definitions are satisfied.

"Linear Space". Show that both addition and scalar multiplication, f+ g and rf, where f and g are in C[a, b] and r is a number. Show that this has all the properties of a vectorspace:
Addition is associative: (f+ g)+ h= f+ (g+ h)
Addition is commutative: f+ g= g+ f
scalar multiplication distributes over addition: r(f+ g)= rf+ rg
r(sf)= (rs)f
There exist a "zero" function so that f+ 0= f for all f in C[a, b]
Every function has a "negative": for each f, there exist g such that f+ g= 0.

Norm: ||f||= sup |f(x)|.

Show that ||f|| is always non-negative and equal to 0 if and only if f is the "zero" function.
Show that ||rf||= |r| ||f|| where f is a function in C[a,b] and r is a number.
Show that $||f+ g||\le ||f||+ ||g||$.