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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: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. ^[ref].

Every normed vector space is a topological vector space: the triangle inequality gives $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$, whilecontrols 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 $x\ne 0$, 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.

The weak topology and norm topology of a separable Banach space generate the same Borel sigma-algebra. A countable norming family makes every norm ball weakly Borel measurable, and separability makes every norm-open set a countable union of such balls. The reverse inclusion follows because the weak topology is coarser.

If the relative weak topology on the unit ball of a normed vector space is first countable at zero, choose a countable collection of finite families of continuous linear functionals defining a local base. For any $f\in E^{\prime }$ and $ϵ>0$, one family $F$ has common kernel $M$ on which $\Vert f{|}_M\Vert \le ϵ$. Extend that restriction to $h\in E^{\prime }$ with $\Vert h\Vert \le ϵ$ using the Hahn-Banach theorem. Then $f-h$ vanishes on $M$, so it belongs to $\mathrm{span}F$ by finite-dimensional factorization. Consequently the countable union of these families has norm-dense linear span in $E^{\prime }$. The converse is the metrization of the weak topology on a bounded set construction from a norm-dense sequence in $E^{\prime }$. In particular, separability of $E$ alone does not suffice: ${\ell }^1$ has nonseparable dual ${\ell }^{\infty }$.

The weak closure ${\overline{K}}^w$ of $K$ is its closure in the weak topology. Thus $x\in {\overline{K}}^w$ exactly when every weak neighbourhood of $x$ meets $K$.

A Banach space is reflexive if and only if its closed unit ball is compact in the weak topology. One direction applies the Banach-Alaoglu theorem to the unit ball in the bidual; the converse follows because Goldstine theorem makes the canonical image of the unit ball weak-star dense, while weak compactness makes that image weak-star closed.

A bounded linear operator between Banach spaces is weakly compact exactly when its Banach-space adjoint is weakly compact. Equivalently, that adjoint is continuous from its domain's weak-star topology to its range's weak topology. For the reverse direction, the bidual condition applied to $T^{\ast }$ forces $T^{\ast \ast \ast }$ to annihilate every functional on $Y^{\ast \ast }$ vanishing on $J_{Y}Y$; the Hahn-Banach theorem then places $T^{\ast \ast }(X^{\ast \ast })$ inside $J_{Y}Y$.

A subset of a Banach space is weakly compact when it is compact in the weak topology; it is relatively weakly compact when its weak closure is weakly compact.

The weak topology of a CW complex is determined by its closed cells: a subset is closed if and only if its inverse image under every characteristic map from a closed cell is closed. A map out of the complex is continuous when its restrictions along those characteristic maps are continuous. This is distinct from the weak topology of a vector space. Taking the product with a locally finite CW complex, such as a closed interval with its finite cell structure, retains the corresponding CW weak topology.