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If I have an inner product space, the hierarchy goes:

Inner product space $\Rightarrow$ normed space $\Rightarrow$ metric space $\Rightarrow$ topological space.

The reverse, however, is not always true. I'm currently struggling with this idea. So, it would help a lot if you could feature some examples like:

  1. a topological vector space that is not a metric vector space
  2. a topological (non-vector) space that is not a metric (non-vector) space
  3. a metric vector space that is not a normed vector space
  4. a metric (non-vector) space that is not a normed vector space
  5. a normed vector space that is not a inner product space
  6. a normed space that is not a vector space (if that does exist)
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    $\begingroup$ The discrete metric is the standard example to break norms. For non-inner product spaces there are plenty of options in infinite dimensional function spaces (such as $L^\infty$) $\endgroup$ – AlexR Jul 2 '15 at 21:23
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  1. a topological vector space that is not a metric space: take $V=C(\Bbb{R})$ where the topology is given by convergence on compact sets. A basis for this topology is given by sets of the form $$U_{K,f,\varepsilon} = \{ g : \sup_K |g-f| < \varepsilon \}$$ where $f \in V$ is continuous, $\varepsilon >0$ is a positive real number, $K \subset \Bbb{R}$ is a compact set.

  2. a topological space that is not a metric space : take any infinite set with the cofinite topology.

  3. I don't know the definition of "metric vector space" (if it exists).

  4. a metric space that is not a normed vector space: take the ball $B(0,1)$ in any normed space

  5. a normed vector space that is not a inner product space: take $L^{\infty}(\Bbb{R})$

  6. a normed space that is not a vector space (if that does exist): it does not exist since by definition every normed space is a vector space with a norm.

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  • $\begingroup$ A metric vector space is a vector space with a metric. Examples of vector spaces with metrics are Euclidean spaces, Hilbert spaces, and Banach spaces. All of these are also normed vector spaces $\endgroup$ – étale-cohomology Jul 4 '17 at 8:43
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    $\begingroup$ @étale-cohomology I see. So, is there an example of metric vector space which is not a normed space? $\endgroup$ – Crostul Jul 4 '17 at 8:56
  • $\begingroup$ That I do not know, and I'd like to! $\endgroup$ – étale-cohomology Jul 4 '17 at 9:24

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