# Cardinality of infinite dimensional vector space

Assume that V is an infinite dimensional vector space. I know that if V is a vector space over a field F, then |V|=max{dimV,|F|}. So if we take V=$$\mathbb{R}$$ and F=$$\mathbb{Q}$$ then |V|>|F| and |V|=dimV (Cardinality of a basis of an infinite-dimensional vector space). Is there any example for the case |V|>dimV and |V|=|F|?

• $\dim V=1{{}}$? – Lord Shark the Unknown Jan 22 at 20:27
• @LordSharktheUnknown The title says infinite dimensional vector spaces. I don't know if it applies to the desired example. We will see. I would suggest $\mathbb R^\mathbb N$ over $\mathbb R$. – Dog_69 Jan 22 at 22:16
• How that works? – uio666 Jan 23 at 6:12
• Take $V=\mathbb{R}^∞$ the Vector space of all sequences of real numbers then $V$ is infinite dimensional vector space over $\mathbb{R}$ and $|V|=|\mathbb{R}|^{dimV}=\mathfrak c=|\mathbb{R}|>dimV$. – Akash Patalwanshi Apr 24 at 4:45

Let $$\ \big( \ell^{\infty} , \Vert \cdot \Vert \big) \$$ be the $$\mathbb{R}$$-vector space of all bounded real sequences with the norm $$\ \displaystyle \Vert (x_n) \Vert = \sup_{n \in \mathbb{N}} | x_n | \$$ and $$P = \big\{ (x_n) \in \ell^{\infty} : (\exists N \in \mathbb{N})(\forall n \in \mathbb{N})(n \geq N \to x_n = 0) \big\}$$ be the subspace of all eventually zero sequences. It is straightforward to show that $$\ \{ e_n \}_{n \in \mathbb{N}} \$$ is a countable (Hamel) basis for $$P$$, where $$\ e_n = ( \delta_{nk} )_{k \in \mathbb{N}} \$$ and $$\ \delta_{nk} \$$ is the Kronecker delta.
In advance note that $$\ \mathbb{R} \ni \alpha \mapsto \alpha \cdot e_1 \in P \$$ is an injection. Then clearly $$\dim_{\mathbb{R}}(P) = \big| \{ e_n \}_{n \in \mathbb{N}} \big| = \aleph_0 < | \mathbb{R} | \leq |P| \ \ \ .$$ So, because $$\ |P| = \max \big\{ \! \dim_{\mathbb{R}}(P) , | \mathbb{R} | \big\} \$$ we are left with $$\ |P| = | \mathbb{R} | \$$.