An infinite family of subspaces Good evening. Can you help me with this please? 
Construct an infinite family of subspaces of a vector space $V$, $W_1,W_2,...,W_n,...$ such that $dim(W_n)=∞$ for all n and also $W_n∩W_m=\{0\} $ if $n ≠ m$.
 A: Let $V$ be the space of functions $\mathbb N \to K$, where $K$ is a field.
Then $V$ is a vector space over $K$ of infinite dimension.
Let $X \subseteq \mathbb N$. Then the set $W(X)$ of functions $\mathbb N \to K$  having support in $X$ is a subspace of $V$ whose dimension is the cardinality of $X$. Having support in $X$ means that the function is zero outside $X$.
Let $X_n = 2^n\mathbb N + 2^{n-1}-1$ and let $W_n=W(X_n)$.
Then the $W_n$ have infinite dimension and are pairwise independent because the $X_n$ are infinite and pairwise disjoint.
(We have $X_{n+1} = 2X_n+1$ and so $X_n$  the set of numbers which have binary representation of the form $x\cdots x 0 1\cdots 1$, where there are $n-1$ digits $1$ at the end. This proves that the $X_n$ are pairwise disjoint.)
A: If $V$ is numerable, pick a basis $B:=\{e_n\}_{n\in \mathbb{N}}$, take $W_0$ as the subspace generated by the elements of $B$ of prime index, $W_1$ as the subspace generated by the elements of $B$ of index twice a prime, $W_2$ the subspace generated by the elements of index thrice a prime bigger than 2, and so on.
