What is the meaning of this symbol $\langle x,y^{(k_j)}\rangle ?$

Asked May 23 '21 at 19:46

Active May 23 '21 at 19:46

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In this answer It is written that

a subsequence $\left\{y^{(k_j)}\right\}$ such that $\langle x,y^{(k_j)}\rangle >\varepsilon$

Here I'm unable to understand this $\langle x,y^{(k_j)}\rangle $ meaning

I know that in inner-product $\langle x,y\rangle= x_1y_1+x_2y_2$ where $x=(x_1,x_2)$ and $y=(y_1,y_2)$

Here Im confused about $\langle x,y^{(k_j)}\rangle $ meaning ?

asked May 23 '21 at 19:46

jasmine

1

Both $x$ and $y^{(k_j)}$ are elements of $\ell^1$, so the angle brackets are the inner product in $\ell^1$. – Jackozee Hakkiuz May 23 '21 at 20:02
Ok @JackozeeHakkiuz that mean $\langle x,y^{(k_j)}\rangle =\sum_{j=1}^{\infty}x_jy^{(k_j)}$ Am i right ? – jasmine May 23 '21 at 20:07
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Yes, that's correct. – Andrew McMillan May 23 '21 at 20:36
thanks @AndrewMcMillan – jasmine May 23 '21 at 20:44
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wouldn't it be a dual pairing. $l^1$ has no inner product. – Martin May 24 '21 at 06:03

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