Let X be a vector space equipped with a norm middot Let V

Let X be a vector space, equipped with a norm || middot ||. Let V be a subspace of X (i.e., a subset of X that is a vector space itself, with the same scalars as X). Suppose that V contains some open ball (open in (X, || middot ||)), centered at 0. Show that V = X. Suppose that V contains an open ball (open in (X, || middot ||)). Show that V = X. (You can think of the above in a very intuitive way: a ball centered at 0 contains all possible directions. If a vector space contains it, then it contains all dilations of all directions, thus it has to be the whole space. Note that you are proving that V degree = 0, for any V strict subspace of X.)

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