The converse is also true : if absolute convergence implies convergence in a normed space, then the space is a Banach space.
22.
This follows from the fact that for every normed space " Y ", separability of the continuous dual implies separability.
23.
I wish to prove that if X is a normed space, then every proper linear subspace V of X has empty interior.
24.
An important special case is the following : for every vector in a normed space, there exists a continuous linear functional on such that
25.
The Goldstine theorem states that the unit ball of a normed space is weakly *-dense in the unit ball of the bidual.
26.
The answer is, " not necessarily "; indeed, every infinite-dimensional normed space admits linear operators that are not closable.
27.
:* X is a normed space \ Longleftrightarrow X is a Banach space \ Longleftrightarrow X ^ \ star is a Smith space;
28.
A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the ultrametric triangle inequality,
29.
This general definition is convenient for defining a spatial median of a finite-dimensional normed space, for example, for distributions without a finite mean.
30.
The normed space " X " is uniformly smooth if and only if tends to 0 as " t " tends to 0.
