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So I'm reading about tensor products and wanting to make sure I understand the notion completely.

I understand that [tex]V^* \otimes W[/tex] is the space of linear functions from [tex]V \text{to} W[/tex]. And since [tex]V^{**} \backsimeq V[/tex], we have that [tex]V \otimes W[/tex] is the space of linear functions from [tex]V^* \text{to} W[/tex].

However, in a paper that I'm reading, it is stated that [tex]V \otimes W[/tex] can be thought of also as [tex](V\otimes W)^*[/tex]. But since [tex]V^* \otimes W^* \backsimeq (V\otimes W)^*, \text{we have that} V\otimes W \backsimeq V^* \otimes W^*[/tex] and this is where my understanding stops. Was there a type or is the previous statement true? Thanks.

I understand that [tex]V^* \otimes W[/tex] is the space of linear functions from [tex]V \text{to} W[/tex]. And since [tex]V^{**} \backsimeq V[/tex], we have that [tex]V \otimes W[/tex] is the space of linear functions from [tex]V^* \text{to} W[/tex].

However, in a paper that I'm reading, it is stated that [tex]V \otimes W[/tex] can be thought of also as [tex](V\otimes W)^*[/tex]. But since [tex]V^* \otimes W^* \backsimeq (V\otimes W)^*, \text{we have that} V\otimes W \backsimeq V^* \otimes W^*[/tex] and this is where my understanding stops. Was there a type or is the previous statement true? Thanks.

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