# Is it possibile to prove that a bounded sequence in a Lp normed space is Cauchy using the dominated convergence theorem?

I'm new to functional analysis and I'm studying the Banach spaces.

I have some examples where to prove that a space is not Banach/complete, the procedure is to take a Cauchy sequence in the space and showing that it does not converge to an element of the space.

While it seems quite easy to prove that a Cauchy sequence does not converge to an element of the space, to me it seems harder to prove that a chosen sequence is actually a Cauchy sequence. I know the definition of Cauchy seq but I don't understand very well how to use it.

Say we have a normed space $$(X,||\cdot||_{L^p})$$, we take a sequence $$\{f_k\}$$ in $$X$$ and we show that it is bounded in the $$L^p$$ norm by some integrable function. Moreover we show that the sequence converges pointwise to a function $$f$$. We have all the hypotesis to use the dominated convergence theorem, which says that $$I=\int_X|f_k-f|$$ goes to $$0$$ as $$k$$ goes to infinity.

The $$L^p$$ spaces contain all the measurable functions whose $$L^p$$ norm is finite, and the $$L^p$$ norm of $$f_k-f$$ is defined as $$J=(\int_X|f_k-f|^p)^{1/p}$$. My guess is that since $$I\rightarrow0$$, then $$J\rightarrow0$$ too as $$k\rightarrow+\infty$$.

So, if the guess it true, does this mean that our sequence is Cauchy, i.e. $$||f_n-f_m||_{L^p}<\epsilon$$ $$\forall\epsilon>0,\forall m,n>N\in\mathbb{N}$$ ?

## 1 Answer

I am not sure if I understand the questions completely, but here are some relevant facts: if $$f_n \to f$$ almost everywhere, $$|f_n| \leq g$$ and $$g \in L^{p}$$ then $$f \in L^{p}$$ and $$\int |f_n-f|^{p} \to 0$$. Any convergent sequence is always Cauchy, so $$\{f_n\}$$ is Cauchy in $$L^{p}$$. [Remark: we have to assume that $$g \in L^{p}$$. If we just assume that $$g$$ is integrable then $$\{f_n\}$$ need not be Cauchy].

• Thanks for the reply, is that a theorem or just facts? Because what you stated (if $f_n\rightarrow f, |f_n|\le g\in L^p$ then $f\in L^p, \int|f_n-f|^p\rightarrow0$) seems a slightly modified version of the dominated convergence theorem. – sound wave Oct 26 '18 at 13:33
• LDCT hypotesis: $f_n:X\rightarrow\mathbb{R}$ sequence of measurable functions, $f_n\rightarrow f$ for every element in $X$, $|f_n|<g$ where $g:X\rightarrow [0,+\infty]$ is a summable function (i.e. $\int |g|<\infty$, i.e. $g\in L^1)$ – sound wave Oct 26 '18 at 13:44
• Ah I've just read that a corollary of the LDCT is the dominated convergence in $L^p$-spaces, which seems exactly what you stated link – sound wave Oct 26 '18 at 13:48