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The following are two definitions from Appendix C (Linear Spaces Review) of Introduction to Laplace Transforms and Fourier Series, Second Edition, by Phil Dyke:

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I'm wondering what the difference is between these two definitions?

And shouldn't they both be $= 0$, since the inner product results in a scalar?

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  • $\begingroup$ I think that are the same thing. The interesting fact is that a system is complete if and only is a topological base for the space $\endgroup$ – Federico Fallucca Jul 3 '18 at 16:04
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    $\begingroup$ The definitions are identical. But what do you mean by "shouldn't they both be =0"? $\endgroup$ – Paul Frost Jul 3 '18 at 17:33
  • $\begingroup$ @PaulFrost Thanks for the clarification. One is $= \mathbf{0}$ and the other is $= 0$. Since we're dealing with inner products, the result should always be the scalar $0$, right? $\endgroup$ – The Pointer Jul 3 '18 at 17:34
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    $\begingroup$ Okay, I see. The boldface 0 is a typo. $\endgroup$ – Paul Frost Jul 3 '18 at 17:36
  • $\begingroup$ @PaulFrost Ok, thanks for the clarification. I'm guessing this was an error, since the definitions are identical and relatively close together in the text. $\endgroup$ – The Pointer Jul 3 '18 at 17:36

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