Inner Product vs Metric

Question

So I know the properties of a metric are:

1. $ {\displaystyle d(x,y)\geq 0}$

2. ${\displaystyle d(x,y)=0\Leftrightarrow x=y} $

3. ${\displaystyle d(x,y)=d(y,x)}$

4. ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$ ( triangle inequality)

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Inner products have analogous properties to 1. and 2.

They have a conjugate symmety, which is analogous to 3. if we ignore complex numbers/spaces

They don't have a triangle inequality, but they do have linearity in the first arugment. So they both have a property that allows us to deal with a third vector/point from the space.

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So, since both metrics and inner product are defined to somehow "measure distance", and have similar properties, I was wondering if a metric is in some sense a broader version of an inner product?

The norm in R$^n$ is both an inner product and a metric, but there are other metrics that can be defined on R$^n$ as well.

And if an inner product is just a metric, why do we bother with inner products in the first place?

Answer 1

The inner product is less general than a metric - any inner product induces a metric by its norm ($d(x,y)=||x-y||=\sqrt{\langle x-y, x-y \rangle}$), but not all metric spaces are a consequence of some inner product. An inner product is richer, since it can also give you concepts like angles, projections, and many other geometric features (even if the space is not Euclidean).

But yes, any inner product space is definitely a metric space, which gives it plenty of interesting properties, especially when it is a complete metric space (complete inner product spaces are called Hilbert spaces, and they are some of the richest structures in mathematics).

Answer 2

An inner product on a vector space is defined as a bilinear, symmetric positive-definite form $\langle \cdot, \cdot \rangle$.

1. Symmetry means that $\langle u, v\rangle = \langle v,u\rangle$,
2. Bi-linear means that $\langle \alpha u+ v, w\rangle = \alpha\langle u,w\rangle + \langle v,w\rangle$ with a similar computation in the right entry, and
3. Positive-definiteness means that $\langle v,v\rangle \geq 0$ and equality holds if and only if $v=0$.

Now, one can realize an inner product as a metric by defining the norm of a vector to be $||v|| = \sqrt{\langle v,v\rangle }$ and then a metric between two points in Euclidean space can be realized as

$$ d(p,q) \;\; =\;\; ||p - q|| \;\; =\;\; \sqrt{\langle p-q, p-q \rangle}. $$

There can be many, many, many different kinds of metrics which one can define on a space (vector space or not), but a vector space lends itself to a special kind of metric that can result from the notion of an inner product.

An example of a metric not arising from an inner product is the discrete metric $d:X\times X \to \{0,1\}$ on an arbitrary set $X$:

$$ d(p,q) \;\; =\;\; \begin{cases} 0, & \text{if} \; p=q \\ 1, & \text{if} \; p \neq q\\ \end{cases} $$

We can see that this notion of distance on an arbitrary set $X$ can't possibly arise from an inner product since $X$ doesn't even have a vector space structure. It also yields the bizarre property that every point is unit distance from every other point.