# Two definitions of tensor product: when are they equivalent? [duplicate]

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understanding of the “tensor product of vector spaces”

Take vector spaces $V, W$ over the field $\mathbb{K}$. I've come across two different definitions of tensor product $V \otimes W$ and was wondering whether they are the same thing.

Definition 1. $V \otimes W$ is the quotient space $F(V\times W) / \sim$ where $F(V\times W)$ is the free vector space over $V \times W$ and $\sim$ denotes the following relations:

$$(v_1+v_2, w)\sim (v_1, w)+(v_2,w), \qquad (v, w_1+w_2)\sim (v, w_1) + (v, w_2),$$ $$(\lambda v, w) \sim (v, \lambda w)\sim \lambda (v, w).$$

In other words $V \otimes W$ is the space of formal linear combinations

$$\sum_{j=1}^n \alpha_j v_j \otimes w_j, \qquad \alpha_j \in \mathbb{K}, v_j \in V, w_j \in W$$

where we explicitly require $\otimes$ to be bilinear.

Definition 2 (When $V, W$ are finite-dimensional.) $V \otimes W$ is the space of mappings from $V^\star\times W^\star$ into $\mathbb{K}$ that are linear in each variable.

Question. Under what circumstances do those two definitions coincide (up to canonical isomorphism)? I guess that this happens if and only if both $V$ and $W$ are finite-dimensional.

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