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This can sound like something trivial, but I keep seeing both of these being used towards the Hilbert Space, to the point that I am confused and not sure anymore. Even Wikipedia says that it's

"a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product",

So, we don't need a norm, we just use an inner product to induce a metric, right? Or we do need the norm? Can we have an inner product that gives us a metric, without inducing a norm?

In some other place in the very same article, Hilbert Space is called a Normed Space, even though it's not clear from its definition.


My problem is, that Normed Vector Spaces with an Inner Product are, in fact, quite rare. In general, you often have Normed Spaces, where inner product doesn't induce the norm.

So, how is this with Hilbert Space? Do an Inner Product Space with a metric must to be a Normed Space? If yes, then I can understand why it's called a Normed Space, but if there can be a Inner Product Space, with a metric coming from inner product, that doesn't induce a norm somewhere in between (is it possible?), then why people call Hilbert Space a Normed Space?


I would be glad, if someone could explain this and bring insight onto my confusion :-)

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    $\begingroup$ the norm is $\|v\|:= \langle v, v \rangle ^{1/2}$ $\endgroup$ – Tim kinsella Jan 19 '18 at 21:41
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    $\begingroup$ The inner product naturally induces a norm, given by $\Vert v \Vert=\sqrt{\langle v, v \rangle }$. That this is in fact a norm can be seen via the Cauchy-Schwarz inequality. So whenever one says "Hilbert space", we usually carry the whole bundle of information: the vector space structure, the inner product, the natural norm (as above), the metric from the norm, the topology from said metric etc (unless explicitly said that we are considering another topology/metric/norm etc on the underlying set). $\endgroup$ – Aloizio Macedo Jan 19 '18 at 21:41
  • $\begingroup$ Thank you. Now I see that I also confused myself, to the point that I even formulated question in a different way that for what I wanted to ask. I started from "Normed Space" and concluded that it doesn't have to be an Inner Product Space, while it's an other way around. Thank you for clarification! $\endgroup$ – Kusavil Jan 19 '18 at 21:50
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An Hilbert spaces is a Complete Inner Product Space and it is also a Banach Space because from the Inner product we can canonically defines the norm: $||x||=\sqrt{\langle x,x\rangle}$.

But note that a vector space can be normed but not equipped with an inner product ( as you can see here). So we can have a complete normed space ( a Banach space) that is not an Hilbert space.

Intuitively: a normed space is a vector space in which the vectors have a length, in an inner product space we have also ''angle'' between vectors, and if any Cauchy sequence of vectors converge, than this space is complete and is called an Hilbert space.

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