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Inner product in vector space

The inner product is an operation that measures the similarity between vectors.  In a general way, the inner product could be defined as an operation of 2 operands, which are elements of a vector space. The result is a scalar in the set of the complex numbers:

\[ \left \langle \cdot, \cdot \right \rangle : V \times V \rightarrow \mathbb{C}  \]

Formal properties

For \(x, y, z \in V\) and \(\alpha \in \mathbb{C}\), the inner product must fulfill the following rules:

To be distributive to vector addition:

\( \left \langle x+y, z \right \rangle = \left \langle x, z \right \rangle + \left \langle y, z \right \rangle \)

Conmutative with conjugate (applies when vectors are complex):

\( \left \langle x,y \right \rangle  = \left \langle y, x \right \rangle^* \)

Distributive respect scalar multiplication:

\(  \left \langle \alpha x, y \right \rangle =  \alpha^* \left \langle x, u \right \rangle \)

\(  \left \langle  x, \alpha y \right \rangle =  \alpha \left \langle x, u \right \rangle \)

The self inner product must be necessarily a real number:

\(  \left \langle  x, x \right \rangle \geq 0 \)

The self inner product can be zero only when the element is the null element:

\( \left \langle x,x \right \rangle = 0 \Leftrightarrow x = 0 \)

Inner product in \(\mathbb{R}^2 \)

The inner product in \( \mathbb{R}^2\) is defined as follows:

\( \left \langle x, y \right \rangle = x_0 y_0 + x_1 y_1 \)

In self inner product represents the squared norm of the vector:

\( \left \langle x, x \right \rangle = x^2_0 + y^2_0 = \left \| x \right \|^2 \)

Inner product in finite length signals

In this case, the inner product is defined as:

\[ \left \langle x ,y \right \rangle = \sum_{n= 0}^{N-1} x^*[n] y[n] \]

Properties of vector spaces

Vector spaces must meet the following rules:
Addition to be commutative:
\( x + y = y + x \)

Addition to be distributive:
\( (x+y)+z = x + (y + z) \)

Scalar multiplication to be distributive with respect to vector addition:
\( \alpha\left(x + y \right) = \alpha x + \alpha y\)

Scalar multiplication to be distributive with respect to vector the addition of field scalars:
\( \left( \alpha + \beta \right) x = \alpha x + \beta y \)

Scalar multiplication to be associative:
\( \alpha\left(\beta x \right) = \left(\alpha \beta \right) x \)

It must exist a null element:
\( \exists 0 \in V \ \ | \ \ x + 0 = 0 + x = x \)

It must exist an inverse element for every element in the vector space:
\( \forall x \in V \exists (-x)\ \ | \ \ x + (-x) = 0\)