# Is Hilbert space a Normed Space or a Inner Product Space? Or it have to be both at the same time?

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 :-)

• the norm is $\|v\|:= \langle v, v \rangle ^{1/2}$ Jan 19, 2018 at 21:41
• 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). Jan 19, 2018 at 21:41
• 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! Jan 19, 2018 at 21:50

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: $||x||=\sqrt{\langle x,x\rangle}$.