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. $||f_n-f_m||_{L^p}<\epsilon$ $\forall\epsilon>0,\forall m,n>N\in\mathbb{N}$ ?
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].