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I show that, in a precise sense, the more general possible definition for a strong vector space is that of a small\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textrm{Vect}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -enriched endofunctor of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textrm{Vect}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    that is right orthogonal for every cardinal\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\lambda $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , to the cokernel of the canonical inclusion of the\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\lambda $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -th copower in the\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\lambda $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -th power of the identity functor: these form the objects for a universal r.c.s.v.s. I call\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\Sigma \\textrm{Vect}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . I show this is equivalent to the category of\n                    <jats:italic>ultrafinite summability spaces<\/jats:italic>\n                    defined independently in Bagayoko et al. (Automorphisms and derivations on algebras endowed with formal infinite sums, 2024). I relate this category to what could be understood to be the obvious category of strong vector spaces\n                    <jats:inline-formula>\n                      <jats:tex-math>$$B\\Sigma \\textrm{Vect}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and to the r.c.s.v.s.\n                    <jats:inline-formula>\n                      <jats:tex-math>$$K\\textrm{TVect}_s$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    of separated linearly topologized spaces that are generated by linearly compact spaces. I analyze the monoidal closed structures on various r.c.s.v.s. induced by the natural one on\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textrm{Ind}{\\text {-}}(\\textrm{Vect}^\\textrm{op})$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . In particular with respect to the problem of closure under the tensor product of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textrm{Ind}{\\text {-}}(\\textrm{Vect}^\\textrm{op})$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Most of the technical results apply to a more general class of orthogonal subcategories of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textrm{Ind}{\\text {-}}(\\textrm{Vect}^\\textrm{op})$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and I work with that generality as it\u2019s cost-free.\n                  <\/jats:p>","DOI":"10.1007\/s10485-025-09844-w","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T17:17:18Z","timestamp":1772644638000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Vector Spaces with Formal Infinite Sums"],"prefix":"10.1007","volume":"34","author":[{"given":"Pietro","family":"Freni","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,3,4]]},"reference":[{"key":"9844_CR1","series-title":"London Mathematical Society Lecture Note Series","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600579","volume-title":"Locally Presentable and Accessible Categories","author":"J Ad\u00e1mek","year":"1994","unstructured":"Ad\u00e1mek, J., Rosick\u00fd, J.: Locally Presentable and Accessible Categories. 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