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Metric space vs normed vector space vs inner product space

TODO examples:

metric space that is not a normed vector space
norm vs metric: a norm gives size of one element. A metric is the distance between two elements. Given a norm in a space with subtraction, we can obtain a distance function: the metric induced by a norm.

Figure 4.
Hierarchy of topological, metric, normed and inner product spaces
. Source.

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