How Hilbert space is a topological space?

Question

I'm struggling with definitions. I go from https://en.wikipedia.org/wiki/Hilbert_space:

A Hilbert space is 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.

reading all links. "induced" links to https://en.wikipedia.org/wiki/Subspace_topology:

a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology[1] (or the relative topology,[1] or the induced topology,[1] or the trace topology).[2]
Definition
Given a topological space ( X , τ ) and a subset S of X , the subspace topology on S is defined by

https://en.wikipedia.org/wiki/Topological_space#Definitions

As this definition of a topology is the most commonly used, the set τ of the open sets is commonly called a topology on X .

Questions:

1. Is definition quoted at the beginning complete for Hilbert space? It looks to me what follows in wiki are explanations of terms which are explained in links of the given definition.

2. Is link for "induced" in the definition correct?

3. if Yes to (1) and (2), how one goes from definitions of topology (via sets) to distance induced by inner product? How is it proven Hilbert space is a topological space (I've found neither https://en.wikipedia.org/wiki/Inner_product_space nor https://en.wikipedia.org/wiki/Complete_metric_space mentioning that in definitions).

Answer 1

Linking the word 'induced' to the suspace topology is unfortunate, because here we are not dealing with that.

Instead the way the inner product 'induces' (in the sense of 'gives rise to') a metric is the following (which is a standard construction in linear algebra):

1. The inner product gives rise to a norm by $\|x\| := \sqrt{\langle x, x \rangle}$

2. Any norm on a vector space gives rise to a distance function (metric) $d$ by $d(x, y) := \|x - y\|$

You can look up the definition of norm on Wikipedia to see what properties it must have so that you can check a) that the metric from 2) does indeed have the three properties of metric and b) that the norm defined in 1) is indeed a norm.

EDIT: this is how you get the metric. This is also explained in the linked answer. The other part of information you seem to be missing is that/how every metric (distance function) on a set turns that set into a topological space. I assume this is explained in the wikipedia article on complete metric spaces you linked to, but in short:

Let $X$ be the set with distance function $d$. For each $x \in X$ and $r \in \mathbb{R}$ we can look at the set $B(x, r) = \{y \in X : d(x, y) < r\}$ of points at distance less than $r$ form $x$. Sets like these are called open balls. The collection of all open balls (so of all radii around all points) form a basis of a topology and hence generate a topology, which is the topology we are talking about here.