What is the hypothesis space used by this AND gate Perceptron?
As far as I understand:

A hypothesis is a model which is capable of predicting outputs from inputs, hence the $x_1 + x_2 - 1.5$ is a hypothesis but not the only one. The whole models that have the same capability are regarded as hypothesis space.

We know that in AND gate:

     x1       x2     output
|---------|--------|--------|
|     0   |    0   |    0   |
|     0   |    1   |    0   |
|     1   |    0   |    0   |
|     1   |    1   |    1   |
|---------------------------|


and we have $w .\cdot x + b$, based on which, either $0$ or $1$ turns out as output.
$$w \cdot x + b$$
$$w_1 \cdot x_1 + w_2 \cdot x_2 + b$$

trying all the inputs in this expression:

$w_1 \cdot 0 + w_2 \cdot 0 + b &lt;= 0$ (because the output should be 0)  $b &lt; 0$

$w_1 \cdot 0 + w_2 \cdot 1 + b &lt;= 0$ ---> $ w_2 + b &lt;= 0$  So $w_1 &lt; |b|$  

$w_1 \cdot 1 + w_2 \cdot 0 + b &lt;= 0$ ---> $ w_1 + b &lt;= 0$  So $w_2 &lt; |b|$

$w_1 \cdot 1 + w_2 \cdot 1 + b &gt; 0$ ---> $ w_1 + w_2 + b &gt; 0$ 



Firstly, we initialize the weights and bias parameters and then if needed, change them. 

Here, since $b &lt; 0$ we set it as $-1$

Since $w_1 &lt; |b|$,  $w_2 &lt; |b|$ and weights are not negative, we set them as 1.
So we would have:

$w_1 \cdot 0 + w_2 \cdot 0 + b = -1 &lt; 0$ is right, returns 0 because is negative.

$w_1 \cdot 0 + w_2 \cdot 1 + b = w_2 + b = 1 - 1 = 0$  wrong, it returns 1 while it should return 0

$w_1 \cdot 1 + w_2 \cdot 0 + b &lt;= 0$ ---> $ w_1 + b &lt;= 0$  So $w_2 &lt; |b|$ wrong, it returns 1 while it should return 0

$w_1 \cdot 1 + w_2 \cdot 1 + b &gt; 0$ ---> $ w_1 + w_2 + b &gt; 0$ 

So we set b for a smaller value like -1.5 (Note), then all the expressions would work appropriately. Hence $x_1 + x_2 - 1.5$ is a hypothesis for this problem. 

Note: we know that in using perceptron algorithm, when reaching at any point that is not following the current model, weights and bias are updated as follows:

w = w + yx
b = b + y


Here, in the source that you referred to, maybe for simplicity they haven't done so and have just found a sample of a plausible model (hypothesis) 

The other hypothesizes should follow the previously mentioned rules, thereby, $x_1 + x_2 - 2$ can also be another hypothesis for this problem, etc.