= Topological vector space
{wiki}
A topological vector space is a <vector space> $E$ over $\mathbb K=\mathbb R$ or $\mathbb C$, equipped with a <topology> such that <vector addition> and <scalar multiplication> are jointly <continuous>:
$$
E\times E\longrightarrow E,\quad (x,y)\longmapsto x+y,
\qquad
\mathbb K\times E\longrightarrow E,\quad (\lambda,x)\longmapsto\lambda x.
$$
The domains carry the <product topology>, and the <scalar> field carries its usual <topology>. Joint <continuity> permits both inputs to vary at once. Here the definition does not impose the <Hausdorff> condition; we state it separately when needed. https://www.math.lmu.de/~philip/publications/lectureNotes/philipPeter_FunctionalAnalysis.pdf[Peter Philip, Functional Analysis, Definition 1.1 and Example 1.2]{ref}.
Every <normed vector space> is a <topological vector space>: the <triangle inequality> gives $\|x+y\|\leq\|x\|+\|y\|$, while
$$
\|\lambda x-\lambda_0x_0\|\leq |\lambda|\,\|x-x_0\|+|\lambda-\lambda_0|\,\|x_0\|
$$
controls <scalar multiplication> near any $(\lambda_0,x_0)$. More generally, a <seminorm topology> supplies many measurements of smallness, rather than one <norm>. Translation by any <vector> is a <homeomorphism>, so a <neighbourhood basis> at zero determines all <neighbourhoods> by translation.
The compatibility matters: on a nonzero real <vector space>, the <discrete topology> makes <vector addition> <continuous>, but not <scalar multiplication>. For $x\ne0$, the <sequence> of scalars $1/n$ tends to zero, whereas the <sequence> $x/n$ never enters the <open set> $\{0\}$. Thus choosing a <topology> arbitrarily need not respect the <linear algebra>.
This framework lets <functional analysis> choose the appropriate notion of <limit of a sequence>. <Norm topology> measures one global error, <weak topology> tests <linear functionals>, and the <Fréchet space> of <smooth functions> controls every order of <derivative>. In each case the operations forming <linear combinations> are <continuous>, making approximation compatible with the underlying equations.
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