Question #259387

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


1
Expert's answer
2021-11-04T09:16:37-0400

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

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


for linear space:

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

f(ax)=af(x)f(ax)=af(x)


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

if

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

then

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


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


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


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