Hilbert vs Inner Product Space

Question

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product.

Also can a metric always be defined by an inner product?

Answer 1

If you have a vector space $X$ with an inner product $\langle \cdot, \cdot \rangle$, this defines a norm $\|\cdot\|$ by $\|x\|=\sqrt{\langle x, x\rangle}$ (it is a good exercise to prove that this is in fact a norm). Similarly, this defines a metric, $d(x,y)=\|x-y\|$ (it is again a good exercise to prove that this is in fact a metric). This is the case for any inner product space, so yes, an inner product always defines a metric. However, not every metric is defined by an inner product!

A sequence of elements $\{x_n\}$ in $X$ is called a Cauchy sequence if $\|x_n-x_m\|\to0$ as $n,m\to\infty$. An inner product space $X$ is called a Hilbert space if it is a complete metric space, i.e. if $\{x_n\}$ is a Cauchy sequence in $X$, then there exists $x\in X$ with $\|x-x_n\|\to0$ as $n\to\infty$.

Answer 2

A Hilbert space is an inner product space that is complete with respect to the norm. Completeness is what differentiates the two.

Not every metric space can be defined by an inner product, for instance the space of continuous functions on $[0,1]$ with the supremum norm as its metric doesn't come from an inner product.

In particular, a Banach space (a complete normed vector space) is a Hilbert space iff its norm satisfies the parallelogram law.