Let $1\leq p <\infty$ and $n\geq 1$ and let $W_p^n[0, 1] = $ the functions $f_[0, 1]\to \Bbb F$ such that $f$ has $n-1$ continuous derivatives, $f^{(n-1)}$ is absolutely continsous, and $f^{(n)}\in L^p[0, 1]$. For $f$ in $W^n_p[0, 1]$, define $$ \|f\| = \sum_{k = 0}^n\left[\int_0^1|f^{(k)}(x)|^pdx\right]^{1/p} $$ Then $W^n_p[0, 1]$ is a Banach space.
hi~!!
i'm a newbie in functional analysis :)
above example is in Conway's "A course in functional analysis"
struggling to figure out why that space is Banach.. but no clear idea is in my head
could you give me a sketch of proof or some hints?
please be my savior ^~^
What you need is completeness of $L^{p}$ and repeated application of the following:
Lemma
Let $f_n$'s be absolutely continuous, $f_n \to f$ in $L^{p}$ and $f_n' \to g$ in $L^{p}$. Then $f$ is absolutely continuous and $g=f'$ almost everywhere.
Proof of the lemma: $f_n(x)-f_n(0)=\int_0^{x} f_n'(t)\, dt$. This implies $f(x)-f(0)=\int_0^{x} g\, dt$ and the lemma follows.