TITLE: Norm of a tensor product of operators
QUESTION [12 upvotes]: I have two Hilbert spaces $H_1$ and $H_2$ which are subspaces of a bigger Hilbert space $H$.
I also have two bounded linear functions $T_1:H_1\rightarrow H$ and $T_2:H_2\rightarrow H$.
I define the tensor product space $F=H_1\otimes H_2$, and a linear function on it $T=(T_1\otimes T_2)$. The vector space $F$ has the induced inner product from the Hilbert spaces:
$$\langle\phi_1\otimes \psi_1, \phi_2 \otimes \psi_2\rangle = \langle\phi_1,\phi_2\rangle\langle\psi_1,\psi_2\rangle$$ 
and therefore an induced norm.
I want to show that $\|T\| =\|T_1\| \cdot\|T_2\|$, but I'm stuck since I cant show that
$\|T\| \leq \|T_1\| \cdot\|T_2\|$ (the other inequality I've already shown).
Can anyone help?

REPLY [3 votes]: Let me add the other direction. Suppose $\varepsilon >0$. Then from the boundedness of operators $T_{1}$ and $T_{2}$ we know that there exist unit vectors in the Hilbert spaces $\psi \in H_{1}$, $\xi \in H_{2}$ such that:
$\left \| T_{1} \psi \right \|> \left \| T_{1} \right \| - \epsilon$, $\left \| T_{2} \xi \right \|> \left \| T_{2} \right \| - \epsilon$
Using this we can use the definition of the tensor product to see that:
$\left \| \left (T_{1}\otimes T_{2}  \right )\left ( \psi\otimes \xi \right ) \right \|_{H_{1}\bigotimes H_{2}}=\left \| \left ( T_{1}\psi\otimes T_{2}\xi \right ) \right \|_{H_{1}\bigotimes H_{2}}=\left \|T_{1}\psi \right \|_{H_{1}}  \left \| T_{2}\xi \right \|_{H_{2}}\geq$
$ \geq\left \| T_{1} \right \|\left \| T_{2} \right \|-\epsilon \left \| T_{1} \right \| - \epsilon \left \| T_{2} \right \| + \epsilon^{2}\geq\left \| T_{1} \right \|\left \| T_{2} \right \|-\epsilon \left \| T_{1} \right \| - \epsilon \left \| T_{2} \right \|$
Sending $\epsilon$ to 0 shows the desired result.