Answer to Question #259387 in Functional Analysis for Jayanng

Question

Show that the space L[a,b] of all square integrable functions on the interval [a,b] is a linear space over a vector field R

Answer 1

A function y(x) is said to be square integrable if

$\displaystyle{\int^{\infin}_{-\infin}}|f(x)|^2dx<\infin$

for linear space:

$(f+g)(x)=f(x)+g(x)$

$f(ax)=af(x)$

for the space L[a,b] of all square integrable functions:

if

$\displaystyle{\int^{\infin}_{-\infin}}|(f+g)(x)|^2dx<\infin$

then

$\displaystyle{\int^{\infin}_{-\infin}}|f(x)|^2dx+\displaystyle{\int^{\infin}_{-\infin}}|g(x)|^2dx<\infin$

$\int f(ax)dx=a\int f(x)dx$

so, this is a linear space over a vector field R