I have a question about this theorem: " The closed unit ball of a normed space X is compact iff X is finite-dimensional."
if I take (R,+,.,Q) R is a infinite dimensional vector space when we take rational numbers set as a field. but if we take Euclid norm. we know from Heine-Borel Theorem the closed unit ball in R is compact. this contadicts the theory.
There must be something wrong with my example, but I want to know what it is?
Your space is not a normed space. A normed space is a vector space over $\mathbb{R}$, or sometimes over $\mathbb{C}$ (which is not that different from $\mathbb{R}$ in this context) with some additional structure. By definition.
But not over $\mathbb{Q}$. That would have to be an explicit assumption, $\mathbb{Q}$ behaves very differently from the other two (topological) fields. For example over $\mathbb{Q}$ no finite dimensional ball is compact, while some infinite dimensional are (but not all).
While $\mathbb{C}$ is quite similar to $\mathbb{R}$ in this context. In fact, $\mathbb{C}$-normed spaces are pretty much $\mathbb{R}$-normed spaces of even dimension.