Actually, every finitely universal Banach space contains " almost-isometric " copies of every finite-dimensional normed space, meaning that can be replaced by for every.
42.
Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional ( " non-complete " ) normed spaces which have countable Hamel bases.
43.
A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm.
44.
Similarly, a normed space is Archimedean if a sum of n terms, each equal to a non-zero vector x, has norm greater than one for sufficiently large n.
45.
This example can be extended into a general theorem about the existence of discontinuous linear maps on any infinite-dimensional normed space ( as long as the codomain is not trivial ).
46.
Actually, the existence of such special moduli of continuity for a uniformly continuous function is always ensured whenever the domain is either a compact, or a convex subset of a normed space.
47.
The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space " X " ( " ? " ).
48.
*PM : every finite dimensional subspace of a normed space is closed, id = 6686-- WP guess : every finite dimensional subspace of a normed space is closed-- Status:
49.
*PM : every finite dimensional subspace of a normed space is closed, id = 6686-- WP guess : every finite dimensional subspace of a normed space is closed-- Status:
50.
Equipped with the metric " ? ", the space of isometry classes of " n "-dimensional normed spaces becomes a compact metric space, called the Banach Mazur compactum.
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