Example

— 1) If $E$ is a left vector space over a discrete topological division ring $\mathbf{K}$, the discrete topology on $E$ is compatible with the vector space structure of $E$ (this is not so if $\mathbf{K}$ is non-discrete and $E$ is not the single point 0). 2) Let $A$ be a topological ring (GT, III, § 6.3) and let $\mathbf{K}$ be a subring of $A$ that is also a division ring and such that the topology induced on $\mathbf{K}$ by that of $A$ is compatible with the division ring structure of $\mathbf{K}$; then the topology of $A$ is compatible with its left vector space structure on $\mathbf{K}$. 3) Let $\mathbf{K}$ be any topological division ring and $I$ an arbitrary set. On the product vector space $K_s^I(A, II, § 1.5)$, the product topology is compatible with the vector space structure (GT, III, § 6.4). Or we can say that the space $K_s^I$ of mappings of $I$ in $\mathbf{K}$ with pointwise or simple convergence topology is a topological vector space on $\mathbf{K}$ (TG, X, p. 4). 4) Let $X$ be a topological space; on the set $E = \mathscr{C}(X; \mathbf{R})$ of finite real-valued continuous functions defined over $X$, the compact convergence topology (GT, X, § 1.3) is compatible with the vector space structure of E on $\mathbf{R}$. For, let $u_0$ be a point of E, let H be a compact subset of X and $\varepsilon$ be an arbitrary strictly positive number. The real-valued function $u_0$ is bounded in H ; let $a = \sup_{t \in H} |u_0(t)|$; if $u$ is any point of E then for all $t \in H$

Hence, if $|\lambda - \lambda_0| \leq \varepsilon$ and $|u(t) - u_0(t)| \leq \varepsilon$ for all $t \in H$, then for $t \in H$, $|\lambda u(t) - \lambda_0 u_0(t)| \leq \varepsilon (\varepsilon + |\lambda_0| + a)$, which shows that the axiom (EVT) is satisfied ; similarly it can be verified that the compact convergence topology is compatible with the additive group structure of E.

On the other hand, if X is not compact, the uniform convergence topology (in X) is not necessarily compatible with the vector space structure of E ; for example if $X = \mathbf{R}$ and if $u_0$ is an unbounded continuous function in $\mathbf{R}$, then the mapping $\lambda \mapsto \lambda u_0$ of $\mathbf{R}$ in E is not continuous in the uniform convergence topology on E.

5) Let E be a vector space of finite dimension $n$ over a topological division ring K ; there exists an isomorphism $u : K_s^n \to E$ of vector K-spaces and moreover, if $v$ is a second isomorphism of $K_s^n$ on E, then we can write $v = u \circ f$, where $f$ is an automorphism of the vector K-space $K_s^n$. Consider, on $K_s^n$, the product topology that is compatible with its vector space structure (Example 3) ; since every linear mapping of $K_s^n$ in itself is continuous for this topology, every automorphism of the vector space $K_s^n$ is bicontinuous. Hence, if we transfer the product topology of $K_s^n$ to E, by means of any isomorphism whatever of $K_s^n$ on E, the topology obtained on E is independent of the particular isomorphism used ; we call it the canonical topology on E ; we shall characterize it differently (I, § 1.3) when K is a non-discrete complete division ring with a valuation. Every linear mapping of E in a topological vector space over K is continuous for the canonical topology on E.

In the same way as in def. 1, a right topological vector space over K, a topological division ring, can be defined ; but every right vector space on K can be considered as a left vector space on the division ring $K^0$ opposite to K (A, II, § 1.1) and the topology of K is compatible with the structure of the division ring $K^0$. For this reason we usually consider only left topological vector spaces ; when we speak of « topological vector space » without qualification, it is to be understood that we refer to a left vector space.

If K' is a sub-division ring of K, and E a topological vector space over K, then it is clear that the topology of E is still compatible with the vector space structure of E relative to K', obtained by restricting the field of scalars to K' ; we say that the topological vector space on K', obtained by this procedure, underlies the topological vector space E on K.

In order that a topological vector space E be Hausdorff, it is necessary and sufficient that for all $x \neq 0$ of E, there exists a neighbourhood of 0 not containing $x$ (GT, III, § 1.2).

Consider a topology, on a vector space E over a topological division ring K, that is compatible with the additive group structure of E. Because of the identity

axiom (EVT) is equivalent to the following system of three axioms.

(EVT$_1'$) For all $x_0 \in E$, the mapping $\lambda \mapsto \lambda x_0$ is continuous at $\lambda = 0$.

(EVT$_2'$) For all $\lambda_0 \in K$, the mapping $x \mapsto \lambda_0 x$ is continuous at $x = 0$.

(EVT$_3'$) The mapping $(\lambda, x) \mapsto \lambda x$ is continuous at $(0, 0)$.

In particular: