---- > [!definition] Definition. ([[TVS direct sum]]) > The [[category]] $\mathsf{TVS}$ of [[topological vector space|topological vector spaces]] has finite [[categorical biproduct|biproducts]]: finite [[categorical coproduct|coproducts]] and [[product topology|products]] exist and agree. > Indeed, if $X$ and $Y$ are two [[topological vector space|TVSs]], then the [[inclusion map|canonical injections]] $X \hookrightarrow X \oplus Y$, $Y \hookrightarrow X \oplus Y$ are [[continuous]] with respect to the [[product topology]] on [[finite direct products and coproducts align in the category of abelian groups|the]] [[direct sum of modules|vector space direct sum]] $X \oplus Y=X \times Y$, as are the [[categorical product|canonical projections]] $X \times Y \twoheadrightarrow X$, $X \times Y \twoheadrightarrow Y$.[^1] > In this case we call $X \oplus Y$ the **(topological) direct sum** of $X$ and $Y$. ^definition > [!equivalence] Characterizing internal topological direct sums. > > Suppose $Y,Z$ are [[linear subspace|linear subspaces]] of a [[topological vector space|TVS]] $X$. The following are equivalent: > - *(Categorical definition above)* $Y \oplus Z=Y+Z$ as a topological direct sum, that is, $Y+Z$ satisfies the [[universal property]] for (co)products in $\mathsf{TVS}$. > - *(Most intuitive)* $Y \oplus Z=Y+Z$ as an *algebraic* direct sum and the addition map $\begin{align} > Y \oplus Z &\xrightarrow{(+)} Y+Z \\ > (y,z) & \mapsto y+z > \end{align}$ is a [[homeomorphism]]. > - *(One way to check)* $Y \cap Z=(0)$ and $(+)^{-1}$ is [[continuous]] > - *(Another way)* $Y \cap Z=(0)$ and one, hence both, of the projections $y+z \xmapsto{p_{Y}} y$ or $y+z \xmapsto{p_{Z}}z$ is [[continuous]]. > > > > > [!proof]- Proof. > > > > We ask: $\text{When does }Y \oplus Z = Y+Z \text{ as a topological direct sum}?$ > > > > In other words, when does $Y+Z$ satisfy the (co)product [[universal property]] in $\mathsf{TVS}$? *Algebraically* [[direct sum of abelian groups|we know it is necessary and sufficient]] that $(+):Y \oplus Z \to Y+Z$ is a [[bijection]]. But *topologically* we require also the projections $G+H \to G$ and $G+H \to H$ to be [[continuous]]; equivalently[^4] for $(+)^{-1}$ to be [[continuous]]. Since $(+)$ is [[continuous]] by the [[topological vector space|TVS]] definition, the three equivalences follow. > [^4]: The [[universal property]] specifies *both* projections to be continuous. This happens if and only if $(+)^{-1}$ is [[continuous]], as follows from the identity $(+ ^{-1}) ^{-1}(U \times V)=p_{Y}^{-1}(U) \cap p_{Z} ^{-1}(V).$ ([[basis for a topology|a]], [[topology generated by a basis|b]], [[product topology|c]], [[characterizing continuity of linear maps|d]]) In our [[topological vector space|TVS]] setting it turns out moreover that $\text{both }p_{Y},p_{Z} \text{ are continuous }\iff \text{ one of }p_{Y},p_{Z} \text{ is continuous}.$This is because $(\iota_{Y+Z}: Y+Z \hookrightarrow X)=\iota_{Y} \circ p_{Y}+ \iota_{Z} \circ p_{Z}.$If (say) $\iota_{Y}$ is [[continuous]], then $\iota_{Z} \circ p_{Z}$ is continuous as the difference of continuous maps. Since $\iota_{Z}$ is an [[topological embedding|embedding]], it follows that $p_{Z}$ is [[continuous]]. [^1]: [[direct sum of modules|We already know]] [[finite direct products and coproducts align in the category of abelian groups|that]] $X \oplus Y$ and $X \times Y$, together with their respective canonical injections and projections, exist and satisfy the same [[universal property]] for (co)products in the category $\mathsf{Vect}_{\mathbb{F}}$. Hence all that must be shown to verify their restriction to the [[subcategory]] $\mathsf{TVS}$ is that the canonical injections and projections belong, i.e. are *toplinear*; linearity we already know so continuity is all there is to check. > [!basicnonexample] > Not true for the infinite case. ([[TODO]]: show that the product of an infinite family $\{ X_{i} \}$ is $\prod_{i=1} X_{i}$ with the initial [[product topology]]), while the coproduct is the (algebraic) direct sum $\bigoplus_{i}X_{i}$ endowed with the final (coproduct) topology, which is generally strictly [[comparable topologies|finer]]...) ^nonexample ---- #### ---- #### 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 > ``` [[direct sum of abelian groups|Recall]] that if $Y,Z$ are [[linear subspace|linear subspaces]] of a [[vector space]] $X$, [[Minkowski sum|then]] $X_{0}=Y+Z$ is a [[linear subspace]] of $X$ and the natural [[surjection]] $(y,z) \xmapsto{T} y+z$ is [[injection|injective]] iff $Y \cap Z=\{ 0 \}$. In this case we write $X_{0}=Y \oplus Z$ and call $X_{0}$ the (internal) (algebraic) direct sum of $Y$ and $Z$. Crucially, this does not guarantee $X_{0}$ to be the If $X$ is moreover a [[topological vector space]], then $T$ is by definition [[continuous]] but $T^{-1}$ may not be: $T^{-1}$ is [[continuous]] iff the projections $Y+Z \to Y$ and $Y+Z \to Z$ are. If $T$ is indeed a [[homeomorphism]] then we write $X_{0}=Y \oplus Z$ and call $X_{0}$ the **(internal) topological direct sum**