Locally convex spaces keep linear topology without forcing a single norm. Seminorms let one control different derivatives, compact sets, or coordinates separately.
Seminorm families
A locally convex topology may need many seminorms rather than one norm. Each seminorm records a different mode of control.
Definition. A seminorm \(p\) on a vector space \(X\) satisfies \(p(x+y)\le p(x)+p(y)\) and \(p(\alpha x)=|\alpha|p(x)\), but may vanish on nonzero vectors. A family \(\mathcal P\) of seminorms separates points if for every \(x\ne0\) some \(p\in\mathcal P\) has \(p(x)>0\).
Proposition. A separating family \(\mathcal P\) defines a Hausdorff locally convex topology whose basic neighborhoods of \(0\) are \(\{x:p_i(x)<\varepsilon,\ i=1,\ldots,n\}\).
Proof. These sets are balanced, convex, and absorbing, and they are stable under finite intersections. Translates define a vector topology because addition and scalar multiplication are controlled by the seminorm inequalities. Hausdorffness follows from separation: if \(x\ne0\), choose \(p(x)>0\), then neighborhoods with \(p(\cdot)<p(x)/2\) separate \(0\) and \(x\).
Continuity of linear maps
Linearity makes continuity a local question at the origin. In normed or locally convex spaces, this local control becomes a usable global estimate.
Theorem. Let \(X\) be defined by seminorms \(\mathcal P\), and let \(Y\) be normed. A linear map \(T:X\to Y\) is continuous iff there exist \(p_1,\ldots,p_n\in\mathcal P\) and \(C<\infty\) such that \(\|Tx\|\le C\max_i p_i(x)\) for all \(x\in X\).
Proof. If the estimate holds, continuity at \(0\) is immediate. Conversely, continuity at \(0\) gives a basic neighborhood \(U=\{x:p_i(x)<\varepsilon,\ i=1,\ldots,n\}\) such that \(T(U)\subset B_Y(0,1)\). For \(x\ne0\), set \(q(x)=\max_i p_i(x)\). If \(q(x)>0\), then \(\varepsilon x/(2q(x))\in U\), so \(\|Tx\|\le2q(x)/\varepsilon\). If \(q(x)=0\), then \(tx\in U\) for every \(t>0\), so \(t\|Tx\|<1\) for all \(t\), forcing \(Tx=0\).
Continuous linear functionals
Continuous linear functionals are the scalar tests compatible with the chosen topology. They determine the weak viewpoint associated with the space.
Corollary. A linear functional \(f:X\to\mathbb F\) is continuous iff it is dominated by a finite maximum of defining seminorms: \(|f(x)|\le C\max_{1\le i\le n}p_i(x)\).
Proof. Apply the preceding theorem with \(Y=\mathbb F\).
Weak topology as locally convex topology
The weak topology is the smallest locally convex topology preserving a chosen family of scalar observations.
Example. The weak topology \(\sigma(X,X^*)\) is generated by seminorms \(p_f(x)=|f(x)|\), \(f\in X^*\). The weak-star topology \(\sigma(X^*,X)\) is generated by seminorms \(p_x(f)=|f(x)|\), \(x\in X\).
Fr?chet spaces
Fr?chet spaces keep completeness and metrizability without requiring a norm. They are the natural setting for many spaces of smooth functions.
Definition. A Fr?chet space is a complete metrizable locally convex space. If \((p_n)\) is a countable separating family of seminorms, a compatible translation-invariant metric is \(d(x,y)=\sum_{n=1}^\infty2^{-n}p_n(x-y)/(1+p_n(x-y))\).
Proposition. Convergence in this metric is exactly convergence in every seminorm \(p_n\).
Proof. If \(d(x_k,x)\to0\), each summand forces \(p_n(x_k-x)\to0\). Conversely, if every seminorm tends to \(0\), first choose \(N\) so \(\sum_{n>N}2^{-n}<\varepsilon/2\), then choose \(k\) so the first \(N\) summands are collectively below \(\varepsilon/2\).
Spaces of test functions
Test-function spaces are locally convex because one must control supports and all derivatives. A single norm cannot encode this structure adequately.
Remark. Distribution theory needs locally convex spaces because \(C_c^\infty(U)\) is not naturally Banach. Its topology records all derivatives on all compact subsets, and continuity of a distribution means estimates against finitely many such seminorms locally.