I'm currently trying to understand my error better in my work for problem 3.2.21 from Pederson's Analysis now, which states the following:
"Let $\mathfrak{H},\mathfrak{K}$ be separable Hilbert spaces and form the usual tensor product $\mathfrak{H} \otimes \mathfrak{K}$. Given $S \in \mathcal{B}(\mathfrak{H})$ and $T \in \mathcal{B}(\mathfrak{K})$, prove there exists a unique operator $S \otimes T \in \mathcal{B}(\mathfrak{H} \otimes \mathfrak{K})$ such that: $$(S \otimes T)(e_n \otimes f_m) = \sum_{k=1}^{\infty} \sum_{j=1}^{\infty} \langle Se_n,e_k \rangle_{\mathfrak{H}} \langle Tf_m, f_j \rangle_{\mathfrak{K}} (e_k \otimes f_j)$$ for all $n,m$ and $(e_k)_{k},(f_j)_j$ orthonormal bases of $\mathfrak{H},\mathfrak{K}$ respectively. Moreover prove that $\lVert S \otimes T \rVert = \lVert S \rVert \lVert T \rVert$."
I ended up defining $S \otimes T$ first on $\text{span}\{e_j \otimes f_k \: | \: j,k \in \mathbb{N}\}$ by $(S \otimes T)(e_k \otimes f_j) = S(e_k) \otimes T(f_j)$ and tried to extend linearly. However I needed to first prove that $S \otimes T$ on this span is a bounded operator, which I thought would be simple enough since we could just use: $$\left\lVert (S \otimes T)\left(\sum_{i=1}^n \alpha_{i}(e_{k_i} \otimes f_{j_i}) \right) \right\rVert^2 = \left\lVert \sum_{i=1}^n \alpha_{i}(S \otimes T)(e_{k_i} \otimes f_{j_i}) \right\rVert^2 $$ $$= \left\lVert \sum_{i=1}^n \alpha_{i}(S(e_{k_i}) \otimes T(f_{j_i})) \right\rVert^2 \leq \sum_{i=1}^n |\alpha_{i}|^2 \lVert S(e_{k_i}) \otimes T(f_{j_i}) \rVert^2$$ $$\leq \lVert S \rVert^2 \lVert T \rVert^2 \sum_{i=1}^n |\alpha_{i}|^2 = \lVert S \rVert^2 \lVert T \rVert^2 \left\lVert \sum_{i=1}^n \alpha_i(e_{k_i} \otimes f_{j_i}) \right\rVert^2$$ so $\lVert S \otimes T \rVert \leq \lVert S \rVert \lVert T \rVert$ on $\text{span}\{e_j \otimes f_k \: | \: j,k \in \mathbb{N}\}$ and thus extends to $\overline{\text{span}\{e_j \otimes f_k \: | \: j,k \in \mathbb{N}\}} = \mathfrak{H} \otimes \mathfrak{K}$ with the norm on the completion still bounded above by $\lVert S \rVert \lVert T \rVert$.
However, my professor said this was incorrect, as the elements in the vector space of all formal linear combinations of the form $\sum_{n,m} \alpha_{nm} e_n \otimes f_m$ don't look like $\sum_{i=1}^n \alpha_{i}(e_{k_i} \otimes f_{j_i})$, which confuses me, since I thought the vector space of all such formal linear combinations of this form was just, well, the set of all linear combinations of the tensors $e_k \otimes f_j$, and thus I've only shown boundedness on a smaller subset than the span.
Can someone help me understand whats going on here and what I did wrong? Thank you.