> If the value of a company is $10m and the company asks an investor to give $10m in exchange for equity, the investor should own 100%.

That's not investing, that's buying. Buying means the buyer gives $10m to the previous owners, at which point as you say, the previous owner owns 0% and the new owner owns 100%. But the company is in the same position as it was before -- the same amount of cash on hand as it did before.

For investing, you're putting cash into the company's account, which raises the total value of the company.

Value of the company before investment: intangibles + pre-investment cash - debt = $10m

Suppose I own 10% pre-investment; 10% of $10m is $1m of estimated value.

Value of company after the investment: intangibles + pre-investment cash - debt + $10m == $20m

Now I own 5% of $20m, which is still $1m of estimated value. The investor owns 50% of $20m, which is still $10m of estimated value.

In practice of course, there are different classes of shares which end up being paid out differently.