I have come across the following definitions while learning about tensor products of vector spaces.
Notation: $ \mathbb{F} $ will denote the field $ \mathbb{R} $ or $ \mathbb{C} $. Given a vector space $ X $, $ X^{\sharp} $ will denote the space of all linear functionals on $ X $. If $ X $ is a normed vector space then $ X^* $ will denote the space of all bounded linear functionals of $ X $.
Definition 1: Let $ X $ and $ Y $ be vector spaces. For each $ x \in X $, $ y \in Y $ define $ x \otimes y $ for all bilinear functionals $ T : X \times Y \to \mathbb{F} $ by:
$$ (x \otimes y)(T) = T(x, y) $$
Definition 2: Let $ X $ and $ Y $ be vector spaces. For each $ x \in X $, $ y \in Y $ define $ x \otimes y $ for all linear functionals $ f \in X^{\sharp} $ and $ g \in Y^{\sharp} $ by:
$$ (x \otimes y)(f, g) = f(x)g(y) $$
In both definitions, $ X \otimes Y $ is defined to be the linear span of the set $ \{ x \otimes y : x \in X, y \in Y \} $.
Question 1: Are these two definitions equivalent and if so, in what way are they equivalent? If not, do these two definitions have different names?
Clearly, if $ f \in X^{\sharp} $ and $ g \in Y^{\sharp} $ then $ T_{f,g} : X \times Y \to \mathbb{F} $ defined by $ T_{f, g}(x, y) = f(x)g(y) $ is a bilinear map. Conversely, given a bilinear map $ T : X \times Y \to \mathbb{F} $, for each fixed $ x_0 \in X $, $ T_{x_0} : Y \to \mathbb{F} $ defined by $ T_{x_0}(y) = T(x_0, y) $ is a linear functional on $ Y $, and for each fixed $ y_0 \in Y $, $ T_{y_0} : X \to \mathbb{F} $ defined by $ T_{y_0}(x) = T(x, y_0) $ is a linear functional on $ X $. So I think that the above two definitions are equivalent - but I can't formally argue why.
I also have the following two definitions for tensor products of normed vector spaces:
Definition 3: Let $ X $ and $ Y $ be normed vector spaces. For each $ x \in X $, $ y \in Y $ define $ x \otimes y $ for all bounded bilinear functionals $ T : X \times Y \to \mathbb{F} $ by:
$$ (x \otimes y)(T) = T(x, y) $$
Where $ T : X \times Y \to \mathbb{F} $ is bounded if:
$$ \| T \| = \sup_{\| x \| \leq 1, \| y \| \leq 1} |T(x, y)| < \infty $$
Definition 4: Let $ X $ and $ Y $ be normed vector spaces. For each $ x \in X $, $ y \in Y $ define $ x \otimes y $ for all $ f \in X^* $, $ g \in Y^* $ by:
$$ (x \otimes y)(f, g) = f(x)g(y) $$
Again, in both definitions $ X \otimes Y $ is defined to be the linear span of $ \{ x \otimes y : x \in X, y \in Y \} $.
Question 2: How do we distinguish between definitions 1&2 and definitions 3&4 when dealing with normed vector spaces $ X $ and $ Y $? What I mean by this is - given two normed vector spaces $ X $ and $ Y $, should we assume that the elementary tensors $ x \otimes y $ are defined on bilinear/linear functionals OR just bounded-bilinear/bounded-linear functionals? ...Or does it not matter?
Edit - $ X $ and $ Y $ are not assumed to be finite-dimensional.