Are normed vector spaces with countably infinite algebraic dimension never complete?
9295
Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.
14516
The space of continuous functions $C([0,1])$ is not complete in the $L^2$ norm
151117
Are all fields vector spaces?
34206
Quadratic form under $\mathbb{C}$ finite-dimension vector space always have a nonzero vector that yields zero?
70543
Does an uncountable discrete subspace of the reals exist?
44738
Compact operators on an infinite dimensional Banach space cannot be surjective
81229
Show that $(C([a,b]),\|{\cdot}\|_2)$ is not a complete normed vector space
129243
Is $C_{c}^{\infty}[a,b]$ a complete metric space?
119671
Is there an infinite dimensional $V$ such that every linear operator on $V$ is a direct sum of a nilpotent and  an invertible operator?
70413
A vector space over $R$ is not a countable union of proper subspaces
1799
How many k-dimensional subspaces there are in n-dimensional vector space over $\mathbb F_p$?
117575
Every proper subspace of a normed vector space has empty interior
151354
Why are there no non-trivial regular maps $\mathbb{P}^n \to \mathbb{P}^m$ when $n > m$?
91173
Finite-dimensional subspace normed vector space is closed
111001
Does weak topology preserve topological vector space structure?
29030
Vector space with new scalar multiplication defined over finite field still vector space?
28685
If every linear operator on $X$ is bounded then is $\dim X<\infty$?