---- > [!definition] Definition. ([[direct sum of Banach spaces]]) > The [[TVS direct sum|topological direct sum]] $X \oplus Y=X \times Y$[^2] of two [[norm|normed spaces]] carries the following [[norm|norms]], called **$L^{p}$-direct sum norms**, that witness it to also be a [[norm|normed space]]: $\|(x,y)\|_{p}=\begin{cases} (\|x\|^{p} + \|y\|^{p})^{1/p} & 1 \leq p < \infty \\ \max \{ \|x\|, \|y\| \} & p = \infty. \end{cases}$ All these [[norm|norms]] are [[norm|equivalent]], and the [[topological space|topology]] they [[metric topology|induce]] is indeed the [[product topology]] on $X \times Y$. As a consequence,[^1] wrt these norms one has $(f_{k}, g_{k}) \to (f,g)$ iff $f_{k} \to f$ and $g_{k} \to g$. > $X \oplus _{p}Y$ is [[Banach space|Banach]] whenever $X$ and $Y$ are [[Banach space|Banach]]. [^1]: Recall the characterization of [[product topology]] as the 'topology of pointwise (e.g. coordinate-wise) convergence'. > [!basicproperties] > - With this norm, a [[sequence]] $(f_{1},g_{2}), (f_{2},g_{2}), \dots$ in $V \oplus W$ [[sequence|converges]] to some $(f,g)$ if and only if $\lim_{k \to \infty} f_{k}=f$ and $\lim_{k \to \infty}g_{k}=g$. ^properties [^2]: Recall that [[categorical biproduct|finite biproducts]] exist in [[category|categories]] of [[vector space|vector spaces]]. So we can write $X \oplus Y=X \times Y$ (equality is taken to meant 'canonically isomorphic'). > [!justification] > **The $L^{p}$-direct sum norms are all equivalent.** First note the following inequality, valid for all $a,b \geq 0$ and $p \in [1, \infty)$ $\max \{ a, b \} \leq (a^{p}+ b ^{p})^{1/p} \leq 2^{1/p} \max \{ a, b \}.$ > Applied to $\|(\cdot , \cdot)\|_{p}$ this inequality implies that given $p \in [1, \infty)$ and $(x,y) \in X \times Y$ we have $c\|(x,y)\|_{\infty} \leq \|(x,y)\|_{p} \leq C \|(x,y)\|_{\infty},$ > where $c=1$ and $C=2^{1/p}$. Thus any norm is equivalent to the $\infty$-norm, and hence to one another. > > > **An $L^{p}$-direct sum norm induces the product topology.** Since [[norm|equivalent norms are topologically equivalent]], it is enough to check for the $L^{\infty}$ direct sum norm; we will do this via the [[basis-nestling characterization of comparing topologies]]. Let $U \times V$ be a [[topology generated by a basis|basic open set]] in the [[product topology]] on $X \times Y$. Since $U$ is open in $X$ and $V$ is open in $Y$, there exists $u_{0} \in U$, $v_{0} \in V$ and $r_{u},r_{v}>0$ such that $B_{r_{u}}(u_{0}) \subset U$ and $B_{r_{v}}(v_{0}) \subset V$. Put $r:=\min \{ r_{u}, r_{v} \}$. Then $B_{r}^{\infty}\big( (u_{0}, v_{0}) \big)=\{ (u, v) \in U \times V : \max\{ \|u-u_{0}\|, \|v-v_{0}\| \}< r\} $is a [[basis for a topology|basic open set]] for the [[metric topology|topology induced by]] $\|(\cdot, \cdot)\|_{\infty}$, and is contained in $U \times V$. Conversely, for any basic open set $B_{r}\big( (p_{0}, q_{0}) \big)$ in the topology on $X \times Y$ induced by $\|(\cdot, \cdot)\|_{\infty}$ we have $B_{r}\big( (p_{0}, q_{0}) \big) \subset B_{r}(p_{0}) \times B_{r}(q_{0}).$ > > > **The spaces $X \oplus_{p} Y$ are complete when $X,Y$ are.** First recall that if $\|\cdot\|_{\alpha}$, $\|\cdot\|_{\beta}$ are two [[norm|equivalent norms]] on $X$, then every $\|\cdot\|_{\alpha}$-Cauchy sequence is also a $\|\cdot\|_{\beta}$-Cauchy sequence and conversely. Thus it suffices to check [[complete|completeness]] for $p=\infty$. So let $(f_{1},g_{1}), (f_{2},g_{2}),\dots$ be a [[sequence]] in $X \times Y$ that is [[Cauchy sequence|Cauchy]] with respect to $\|(\cdot , \cdot)\|_{\infty}$. Given $\varepsilon>0$ and $i,j \in \mathbb{N}$, observe $\begin{align} > \|(f_{k}, g_{k})-(f_{j}, g_{j})\| _{\infty} < \varepsilon & \iff \max \{ \|f_{k}-f_{j}\|, \|g_{k}- g_{j}\| \} < \varepsilon \\ > & \iff \|f_{k}-f_{j}\|< \varepsilon \text{ and } \|g_k-g_{j}\|<\varepsilon . > \end{align}$ > It follows that the coordinate sequences $(f_{k}) \in X^{\mathbb{N}}$ and $(g_{k}) \in Y^{\mathbb{N}}$ are both Cauchy; since $X$ and $Y$ are [[complete]] we have $f_{k} \to f$ for some $f \in X$ and $g_{k} \to g$ for some $g \in Y$. The result follows since the [[product topology]] is the [[product topology|topology of pointwise convergence]]. > ^justification ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```