# How to show that the sum of $L^p$ spaces is Banach.

Let $p<q$ be positive integers (with the allowance that $q$ may be $\infty$). How can we show that the sum of $L^p$ and $L^q$ is a Banach space under the norm $\|f\|=\inf\{\|g\|_p+\|h\|_q: g+h=f\}$?

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.

Hint: for each $n$, choose $g_n \in L^p$, $h_n \in L^q$ such that $f_n = g_n + h_n$ and $||g_n||_p + ||h_n||_q \le ||f_n|| + 2^{-n}$.
Let $$(X_1,||\cdot||_1)$$ and $$(X_2,||\cdot||_2)$$ two Banach spaces such that $$X_i\subset V$$ where $$V$$ 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 $$\sum_{k=0}^{+\infty}y_1^{(k)}+\sum_{k=0}^{+\infty}y_2^{(k)}+y^{(0)}$$.
• Could you please explain why $\|\cdot\|_{X}$ is a norm on $X$?