Have you ever thought, if only I could extract gold from the sea? maybe I would be rich?

Well, if it was economical someone else would have done it by now, so lets have a look at what is in seawater.

The main constituent of seawater is.. you got it.. water, but lets ignore water for now, and concentrate on the salts in seawater which represent 3.5% of the seawater (by weight)

The main elements are:
Chlorine 1.94%
Sodium 1.08%
Magnesium 0.13%
Sulphur 0.09%
Others 0.26%

there is no surprise there that sodium and chlorine are the main constituents, as common table salt is sodium chloride.
there is also history of magnesium extraction from seawater as well.

However once the abundance becomes lower than magnesium the economics of extraction doesn't start working out.

So how much gold is in seawater?

Estimates put it at 1 part per 100 Trillion. That means to mine 1 gram of it ( about $50) you would need to process 100 Gigalitres of seawater.

I am guessing that processing 100 Gigalitres of seawater is more expensive than $50, hence why we haven't seen this done before?

But what about other heavy metals?

There are some places around the world that have unfortunately polluted their waterways, and there is in theory an economic source of extracting heavy metals that have settled on the riverbank (like alluvial gold does). I hope these projects get funded eventually as they have some economic benefit and an environmental benefit too as it helps clean up polluted waterways.