Assume that $V$ is an infinite dimensional vector space. I know that if $V$ is a vector space over a field F, then $|V|=\max\{\text{dim}V,|F|\}$. So if we take $V=\mathbb{R}$ and $F=\mathbb{Q}$ then $|V|>|F|$ and $|V|=\text{dim}V$ (Cardinality of a basis of an infinite-dimensional vector space). Is there any example for the case $|V|>\text{dim}V$ and $|V|=|F|$?
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$\dim V=1{{}}$? – Angina Seng Jan 22 '19 at 20:27
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@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 '19 at 22:16
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How that works? – uio666 Jan 23 '19 at 06:12
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1Take $V=\mathbb{R}^∞$ 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 '19 at 04:45
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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} | \ $.
Gustavo
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