# What Is the Difference Between These Two Definitions? Infinite Orthonormal Systems and Completeness

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

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?

• 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 – Federico Fallucca Jul 3 '18 at 16:04
• The definitions are identical. But what do you mean by "shouldn't they both be =0"? – Paul Frost Jul 3 '18 at 17:33
• @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? – The Pointer Jul 3 '18 at 17:34
• Okay, I see. The boldface 0 is a typo. – Paul Frost Jul 3 '18 at 17:36
• @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. – The Pointer Jul 3 '18 at 17:36