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Let $\mathsf{TopVect}$ be the category of TVS over $\mathbb K\in \{\mathbb R, \mathbb C\}$ with continuous linear maps as morphisms.

Do the normed spaces or locally convex spaces form a reflective subcategory of $\mathsf{TopVect}$? (in the latter case: do normed spaces form a reflective subcategory of the locally convex spaces?).

Basically: can you "complete" a TVS to a locally convex / normed space?

I have virtually zero intution regarding this topic, but answering these questions might help with this question.

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Without having checked all details I believe that LCS is reflective in TVS but NORM is not:

Given a TVS $(X,\scr T)$ let $\scr T^{lcs}$ be the locally convex topology having the $0$-neighbourhood basis $\lbrace \Gamma(U): U \in {\scr U}_0(X,\scr T)\rbrace$ where $\Gamma(U)=\lbrace \sum \lambda_j u_j: u_j\in U, \sum|\lambda_j|\le 1\rbrace$ denotes the absolutely convex hull. The identity $(X,{\scr T}) \to (X,\scr T^{lcs})$ is continuous and for every continuous linear $f:(X,{\scr T})\to (Y,\scr S)$ with values in a locally convex space, $f$ is also continuous with respect to the associated locally convex topology.

NORM is not reflective (neither in TVS nor in LCS): Consider any non-normed TVS $(X,\scr T)$, e.g. $X=\mathbb R^\mathbb N$ with the product topology, and assume that there is a morphism $f:X\to Y$ into a normed space such that $id:X\to X$ factorizes as $id=g\circ f$. Then $X$ has convex bounded $0$-neighbourhoods (and is thus normed by the Minkowski functional): For the unit ball $B$ of $Y$ the image $g(B)$ is bounded and because of the continuity of $f$ there is a convex $0$-neighbourhood $U$ in $X$ with $f(U)\subseteq B$, hence $U=g(f(U))$ is bounded.

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