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Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in W$,

(1) the norm is symmetric, in the sense that $\|v\otimes w \|_{V\otimes W}= \|w\otimes v\|\\\\$.

(2) the norm satisfies: $\|v\otimes w \|_{V\otimes W}\leq \|v\|_{W} \cdot \|w\|_V$.

Is it true that the tensor product space ($V\otimes W$, $\|\cdot\|_{V\otimes W}$) will thus be complete (and thus Banach)?

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    $\begingroup$ Let's assume both V and W are infinite dimensional. Injective tensor product and projective tensor product are the very first examples that comes to mind for which $V\otimes W$ is not complete. $\endgroup$
    – Onur Oktay
    Commented Aug 23, 2022 at 14:31
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    $\begingroup$ In fact, it's hard to think of any examples where the algebraic tensor product $V\otimes W$ is complete. Conjecture: if $V$ and $W$ are both infinite dimensional, then $V\otimes W$ is not complete. $\endgroup$
    – Nik Weaver
    Commented Aug 23, 2022 at 14:38
  • $\begingroup$ Good idea ... is the set of rank $\leq r$ tensors closed? I guess this would do it. $\endgroup$
    – Nik Weaver
    Commented Aug 23, 2022 at 16:21
  • $\begingroup$ A suggestion to show that the result is false if both spaces are infinite dimensional. One can find norm one linearly independent sequences $(x_n)$ and $(y_n)$. The series $\Sigma \frac 1 {n^2}x_n \otimes y_n$ converges in the completion but the limit is not in the algebraic tensor product. $\endgroup$
    – terceira
    Commented Aug 23, 2022 at 18:41
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    $\begingroup$ @terceira it sure doesn't look like it could be in the algebraic tensor product, but can you prove this? $\endgroup$
    – Nik Weaver
    Commented Aug 23, 2022 at 19:29

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If $V$ or $W$ finite-dimensional, then $V \otimes W$ consists of sums $$\sum_{j=1}^n v_j\otimes w_j\tag1$$ with a fixed number $n$ of terms.

But when $V,W$ are infinite-dimensional, you have sums $\sum_{j=1}^n v_j\otimes w_j$ with unbounded $n$.

Surely partial sums of $\sum_{j=1}^\infty v_j\otimes w_j$ would be Cauchy in $V \otimes W$ provided $\sum \|v_j\|_V\;\|w_j\|_W < \infty$, but there is no reason to think such a sum would be expressed as a finite sum $(1)$.

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