First of all, for fβC([a;b]) we see that
β£β£fβ£β£2β=(β«abββ£f(x)β£2dx)1/2
is well defined and is in [0;+β) .
Now let's verify the norm properties:
- Separation, β£β£fβ£β£2β=0βf=0 . If f=0,β£β£fβ£β£2β=0 is obvious. Now suppose fξ =0,β£fβ£2ξ =0 is a continuous positive function and thus β«abββ£f(x)β£2dxξ =0, therefore β£β£fβ£β£2βξ =0 .
- Linearity, β£β£Ξ±fβ£β£2β=β£Ξ±β£β
β£β£fβ£β£2β . It follows from linearity of an integral : (β«abββ£Ξ±fβ£2(x)dx)1/2=(β£Ξ±β£2β«abββ£f(x)β£2dx)1/2=β£Ξ±β£β
β£β£fβ£β£2β
- Triangle inequality, β£β£f+gβ£β£2ββ€β£β£fβ£β£2β+β£β£gβ£β£2β .
β«abββ£f+gβ£2(x)dxβ€β«abββ£fβ£2(x)dx+β«abββ£gβ£2(x)dx+2β«abββ£fgβ£(x)dx
We have (β«abββ£fgβ£(x)dx)2β€(β«abββ£f(x)β£2dx)β
(β«abββ£g(x)β£2dx), as
(β«abββ£f(x)β£2dx)1/2β
(β«abββ£g(x)β£2dx)1/2β£fgβ£ββ€2β«abββ£f(x)β£2dxβ£fβ£2β+β«abββ£g(x)β£2dxβ£gβ£2ββ (by applying 2β£abβ£β€a2+b2 to (β«abββ£f(x)β£2dx)1/2β£fβ£β and (β«abββ£g(x)β£2dx)1/2β£gβ£β ) and thus by integrating on [a;b] we find :
(β«abββ£f(x)β£2dx)1/2β
(β«abββ£g(x)β£2dx)1/2β«abββ£fgβ£(x)dxββ€1
Therefore (β«abββ£fgβ£(x)dx)2β€(β«abββ£fβ£2dx)β
(β«abββ£gβ£2dx) .
And we conclude :
β£β£f+gβ£β£22β=β«abββ£f+gβ£2β€β«abββ£fβ£2+β«abββ£gβ£2+2β«abββ£fgβ£
β£β£f+gβ£β£22ββ€β«abββ£fβ£2+β«abββ£gβ£2+2(β«abββ£fβ£2)1/2(β«abββ£gβ£2)1/2
β£β£f+gβ£β£22ββ€(β£β£fβ£β£2β+β£β£gβ£β£2β)2
And thus β£β£f+gβ£β£2ββ€β£β£fβ£β£2β+β£β£gβ£β£2β .
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