Let $p Let $\{f_n\}$ be a sequence in $L^p+L^q$, such that $$\sum_{n=1}^\infty \|f_n\|<\infty.$$ We would like to conclude that this implies that $\sum_{n=1}^\infty f_n$ converges to something in $L^p+L^q$, at which point it follows that $L^p+L^q$ is complete by a basic theorem. We could easily do this if for each $f_i$ it were possible to express $f_i=g_i+h_i$, where $g_i\in L^p, h_i\in L^q$ and $\|f_i\|=\|g_i\|_p+\|h_i\|_q$. Though we know Cauchy sequences in $L^p$ converge, it is not clear that for a given $f$, all sequences (or some sequence) $\{g_n+h_n\}$ such that the $(\|g_n\|_p+\|h_n\|_q)\to \|f_n\|$ have the property that $\{g_n\},\{h_n\}$ converge in their respective spaces. It seems possible to imagine the sum converging without the summands converging in their respective spaces. Any help would be much appreciated.
How to show that the sum of $L^p$ spaces is Banach.
2 Answers
You want in fact to show the following result:
Let $(X_1,||\cdot||_1)$ and $(X_2,||\cdot||_2)$ two Banach spaces such that $X_i\subset V$ where $f$ is a vector space. We define $X=\{x_1+x_2,x_1\in X_1,x_2\in X_2\}$ endowed with the norm $||x||_X:=\inf\{||x_1||_1+||x_2||_2,x=x_1+x_2\}$. Then $(X,||\cdot||_X)$ is a Banach space.
To see that, take $\{x^{(n)}\}$ a Cauchy sequence in $X$. We can extract a subsequence, denoted $\{y^{(k)}\}$ such that $||y^{(k+1)}-y^{(k)}||_X\leq 2^{-k}$ for all $k$. Let $(y_1^{(k)}, y_2^{(k)})\in X_1\times X_2$ such that $||y^{(k+1)}-y^{(k)}||_X+2^{-k}\geq ||y_1^{(k)}||_{X_1}+||y_2^{(k)}||_{X_2}$ and $y^{(k+1)}-y^{(k)}=y_1^{(k)}+y_2^{(k)}$. Since $X_1$ and $X_2$ are Banach spaces we can define $y_1:=\sum_{k=0}^{+\infty}y_1^{(k)}$ and $y_2:=\sum_{k=0}^{+\infty}y_2^{(k)}$. We have $$y^{(n+1)}=y^{(0)}+\sum_{k=0}^ny^{(k+1)}-y^{(k)}+y^{(0)}=\sum_{k=1}^ny_1^{(k)}+\sum_{k=1}^ny_2^{(k)}+y^{(0)},$$ which shows that $y^{(n)}$ converges to $\sum_{k=0}^{+\infty}y_1^{(k)}+\sum_{k=0}^{+\infty}y_2^{(k)}+y^{(0)}$.
Hint: for each $n$, choose $g_n \in L^p$, $h_n \in L^q$ such that $f_n = g_n + h_n$ and $||g_n||_p + ||h_n||_q \le ||f_n|| + 2^{-n}$.