I am looking the sequence spaces $l^1$ and $$\{(x_k)_k: \|x\|_{sq}^2 := \sum_{k=1}^\infty k^2\cdot x_k^2 < \infty\}. $$

I am particularly interested in relations between their respective norms: It is fairly easy to show that $\|\cdot \|_{sq}$ is not bounded above by $\|\cdot\|_1$, for example $x_k^{(n)} := k^{-1/2}\cdot \delta_{k,n}$, i.e. $x^{(n)} = (0,0,\ldots,0,n^{-1/2},0,\ldots)$. Then $\|x^{(n)}\|_1\to 0$ but $\|x^{(n)}\|_{sq} \to \infty$.

I am struggling with the other direction. I can find neither a proof that $\|\cdot\|_1$ is bounded above by $\|\cdot \|_{sq}$, neither can I construct a counterexample. I believe that the former is actually true.


Nervermind, Cauchy-Schwarz inequality:

$$\sum_{k=1}^n |x_k^{(n)}| = \sum_{k=1}^n |x_k^{(n)}|\cdot k \cdot \frac{1}{k} \leq \sqrt{\sum_{k=1}^n |x_k^{(n)}|^2\cdot k^2} \cdot \sqrt{\sum_{k=1}^n \frac{1}{k^2}}$$


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