= Fréchet space
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A Fréchet space is a <Hausdorff> <locally convex space> whose <topology> is induced by a complete translation-invariant <metric>. Translation invariance means $d(x+z,y+z)=d(x,y)$; <completeness> means every <Cauchy sequence> converges in the space. Use a translation-invariant <metric> for this condition: an arbitrary compatible <metric> can change which <sequences> are <Cauchy sequences>. https://www.math.lmu.de/~philip/publications/lectureNotes/philipPeter_FunctionalAnalysis.pdf[Peter Philip, Functional Analysis, Definition 1.22 and Remark 1.23]{ref}.
Equivalently, the <topology> comes from a countable <separating family of seminorms> $(p_k)_{k\geq0}$ and is complete in their common sense: if $p_k(x_n-x_m)\to0$ as $n,m\to\infty$ for every fixed $k$, then some $x\in E$ satisfies $p_k(x_n-x)\to0$ for every $k$. A useful compatible translation-invariant <metric> is
$$
d(x,y)=\sum_{k=0}^{\infty}2^{-(k+1)}\min\{1,p_k(x-y)\}.
$$
It is finite because the weights sum to one. The separating condition makes it a <metric>; without that condition the formula gives only a <pseudometric>. Its <convergence in a metric space> and <Cauchy sequence> conditions are exactly the conditions for all the $p_k$ separately: control finitely many terms, then bound the remaining tail by $2^{-(K+1)}$. Replacing $p_k$ by $\max_{0\leq j\leq k}p_j$ gives an increasing family with the same <topology>.
Every <Banach space> is a <Fréchet space>, using its <norm> as the single measurement. The <space of smooth functions on a compact interval> is a <Fréchet space> with <derivative seminorms> but with no single <norm> inducing its smooth <topology>. This is why <Fréchet spaces> are useful in <differential equations>: one can require accuracy in every order of <derivative>, while each particular estimate uses only finitely many orders.
The name honors <Maurice Fréchet>. His doctoral thesis was published as https://link.springer.com/article/10.1007/BF03018603[Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo 22, pages 1–72 (1906)]{ref}; an https://zenodo.org/records/1428464[open scan of Fréchet's original paper]{ref} is available. This identifies a historical source without attributing the modern locally convex definition to that paper.
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