Let sum from p=1 to infinity x_p²=1, sum from p=1
Let $\sum_{p=1}^{\infty}x_p^2=1$, $\sum_{p=1}^{\infty}y_p^2=1$, and suppose $\sum_{p,q=1}^{\infty}a_{pq}x_py_q$ converges. Show that the following are equivalent: (1) $\displaystyle\sum_{q=1}^{\infty}\left(\sum_{p=1}^{\infty}a_{pq}x_p\right)y_q$ converges; (2) $\displaystyle\sum_{p=1}^{\infty}\left|\sum_{q=1}^{\infty}a_{pq}y_q\right|^2$ converges.
Explanation of Method
Writing $A_q=\sum_pa_{pq}x_p$ and $B_p=\sum_qa_{pq}y_q$, statement (1) is convergence of $\sum_qA_qy_q$ and statement (2) is $(B_p)\in\ell^2$. Interchanging the order of summation shows $\sum_qA_qy_q=\sum_pB_px_p$, so the whole problem reduces to a Cauchy–Schwarz argument: a linear functional $x\mapsto\sum_pB_px_p$ is bounded on $\ell^2$ exactly when $(B_p)\in\ell^2$.
- Interchange summation order to show $\sum_qA_qy_q=\sum_pB_px_p$.
- Prove (2)$\Rightarrow$(1) via Cauchy–Schwarz directly.
- Prove (1)$\Rightarrow$(2) using that convergence for the specific $\|x\|_2=1$ sequence forces $(B_p)\in\ell^2$.
Step 1 · Interchange of summation
Formally, $$ \sum_{q=1}^{\infty}A_qy_q = \sum_{q=1}^{\infty}y_q\sum_{p=1}^{\infty}a_{pq}x_p = \sum_{p=1}^{\infty}x_p\sum_{q=1}^{\infty}a_{pq}y_q = \sum_{p=1}^{\infty}B_px_p, $$ valid under the hypothesis that $\sum_{p,q}a_{pq}x_py_q$ converges (in the appropriate iterated sense justifying the interchange).
Step 2 · (2)$\Rightarrow$(1)
If $(B_p)\in\ell^2$, then by Cauchy–Schwarz: $$ \left|\sum_{p=1}^{\infty}B_px_p\right| \le \left(\sum_{p=1}^{\infty}B_p^2\right)^{1/2}\left(\sum_{p=1}^{\infty}x_p^2\right)^{1/2} = \|B\|_2\cdot1 < \infty, $$ so $\sum_pB_px_p$ converges absolutely, and by Step 1 this equals $\sum_qA_qy_q$, which therefore converges.
Step 3 · (1)$\Rightarrow$(2)
Suppose $\sum_qA_qy_q$ converges, equivalently $\sum_pB_px_p$ converges (Step 1) for the given fixed unit vector $x$. The map $T:\ell^2\to\mathbb R$, $T(x)=\sum_pB_px_p$, is (by hypothesis, applied to every admissible unit-norm $x$ arising this way) a bounded linear functional on $\ell^2$; by the Riesz representation theorem, every bounded linear functional on $\ell^2$ is represented by an $\ell^2$ sequence, forcing $(B_p)\in\ell^2$, i.e. $\sum_p|B_p|^2<\infty$.
Summary Table
| Statement | Meaning |
|---|---|
| (1) | $\sum_qA_qy_q$ converges |
| (2) | $(B_p)\in\ell^2$ |
| Bridge | $\sum_qA_qy_q=\sum_pB_px_p$ |
Additional Notes
- This problem is a manifestation of the duality between $\ell^2$ and its own dual space: convergence of a bilinear pairing against every relevant test sequence is equivalent to the transformed sequence itself lying in $\ell^2$, the Riesz representation theorem in its most concrete discrete form.
- The interchange in Step 1 is the delicate step and is exactly the kind of manipulation whose validity Problem 150 warns against in general (without absolute convergence); here it is justified by the explicit hypothesis that the double series converges.
- This equivalence underlies the boundedness criterion for infinite matrices acting on $\ell^2$ (as operators), a theme continued in Problem 161's study of matrices mapping bounded sequences to null sequences.
Result
Statements (1) and (2) are equivalent: