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Tag Archives: metric-spaces
Density of diagonalizable matrices
Consider regarded as a metric space (for instance, identifying it with ). We will prove that the set of diagonalizable matrices is dense in . Let . We want to find a sequence of diagonalizable matrices such that . Suppose … Continue reading
Dense subsets of a metric space sometimes determine its cardinality
Suppose is a metric space and is a dense subset of such that . Then, for every we can produce a sequence of elements in such that . This in turn defines a mapping , which is injective (notice that … Continue reading
An application of Baire’s category theorem
We’ll prove the following fact: if is a continuous function such that, for all we have that as , then as . Let and consider the sets . These sets are closed since is continuous, and we have that , … Continue reading
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