# How to prove that $W^n_p[0,1]$ is a Banach space? [duplicate]

Problem. Let $$1\le p<\infty$$ and $$n \ge 1$$ and let $$W^n_p[0,1]=$$ the functions $$f:[0,1]\to \Bbb{C}$$ such that $$f$$ has $$n-1$$ continuous derivatives, $$f^{(n-1)}$$ is absolutely continuous, and $$f^{(n)} \in L^p[0,1]$$. For $$f$$ in $$W^n_p[0,1]$$, define $$||f||:=\sum_{k=0}^n\left[\int_0^1|f^{(k)}(x)|^p ~\mathrm{d}x\right]^\frac{1}{p}$$ Then $$W^n_p[0,1]$$ is a Banach space.

If we take a Cauchy sequence $$(f_j)_j$$ in $$W^n_p[0,1]$$, then it follows that $$(f_j^{(k)})_j$$ is Cauchy in $$L^p[0,1]$$ for all $$k = 0, 1, \dots, n$$. Hence there are $$g_{k} \in L^p[0,1]$$ such that $$(f_j^{(k)})\to g_k$$ for $$k = 0, 1, \dots, n$$. From this how to prove that $$g_k=g_0^{(k)}$$ for all $$k = 1, \dots, n$$? Thanks.

EDIT I tried to proceed as follows: $$f_j \to g_0$$ in $$L^p$$ $$\implies \text{for some subsequence }f_{j_l}\to g_0$$ a.e. on $$[0,1]$$. Then we have $$g_0(x)= \lim f_{j_l}(x)= \lim \int_0^x f_{j_l}^{'}=\int_0^x\lim f_{j_l}^{'}=\int_0^x g_1$$ for a.e $$x$$ in $$[0, 1]$$. This implies $$g_0'(x) = g_1(x)$$ a.e. $$x$$ . Now similarly we can prove others. Then since $$f_j^{(n-1)}$$ was absolutely continuous functions with $$L^p$$ derivatives we can use this to prove that $$g_{n-1}(=g_0^{(n-1)})$$ is also absolutely continuous with $$L^p$$ derivative. And this shows that $$g_0 \in W^n_p[0,1]$$.

Is there any mistake?

• Does this answer your question? Space Sobolev $W^{m,p}$ complete Commented May 13, 2021 at 6:04
• @lc2r43 No. Actually I don't know "weak derivatives". Commented May 13, 2021 at 6:07
• How do you define the derivative $f^{(n)}$ if not as weak derivative?
– daw
Commented May 14, 2021 at 11:33
• @daw I would define it as : $f^{(n)}(x):=\lim_{h \to 0}\frac{f^{(n-1)}(x+h)-f^{(n-1)}(x)}{h}$ for all $x$. Here $f^{(n-1)}$ is given to be differentiable with a $L^p$ derivative. Right?? Commented May 15, 2021 at 3:20
• No, that limit is not guaranteed to exist, and definitely not for all x, you should learn the definition of a weak derivative before coming back to this question Commented May 21, 2021 at 0:23