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I'm trying to understand this problem. Let's take an infinite dimensional vector space, say [itex]\mathbb{R}^2[/itex] and let [itex]n = 4[/itex]. This problem states we can find a subspace $U$ such that dim(\mathbb{R}^2/U) = 4$. Well, one subspace of $\mathbb{R}^2$ is [itex]U = \{(x, y) : x*y \geq 0\}[/itex] (i.e. the first and third quadrant). So [itex]\mathbb{R}^2 / U = \{v + U : v \in \mathbb{R}^2\}[/itex]. But that means [itex]\mathbb{R}^2 / U = \mathbb{R}^2[/itex], right?