Understanding the notation $H:V\times V\rightarrow \mathbb{F}$ for bilinear forms

Question

I would like to present here definition of Bilinear forms

Here I would like to understand notation

Let $H:V\times V\rightarrow \mathbb{F}$

Does notation $V\times V$ mean Power set of Vector space?i mean Cartesian product? because Cartesian product is defined as

For instance like this

in other words, is Bilinear form linear map from Power set of vector space to some field $F$ ?thanks in advance

Answer 1

The notation $H:V\times V\rightarrow \mathbb{F}$ means that $H$ is a map from the vector space $V \times V$ to the vector space $\Bbb F$.

You're asking: what is $V \times V$ ? As a set, it is the cartesian product of $V$ with itself: $$V \times V = \{(v,w) \mid v,w \in V\}$$

It is moreover $\Bbb F$-vector space with the following componentwise operations: $$(v,w) \oplus (v',w') := (v+v',w+w') \qquad \lambda \odot(v,w):=(\lambda \cdot v,\lambda\cdot w)$$ for any $\lambda \in \Bbb F,v,w\in V$.