Cauchy sequence of vectors when dotted with another vector gives a Cauchy sequence of scalars?

Question

My question is related to vector spaces with an inner product defined (the space is not necessarily complete i.e. not a Hilbert Space)

So imagine I have a Cauchy sequence of vectors $\mathbf{\{x_n}\}$ in this vector space. And I have another arbitrary vector $\mathbf{\Phi}$ in the vector space.

If I have a sequence of scalars defined as follows: $$\xi_n=\langle\mathbf{x_n},\mathbf{\Phi}\rangle$$

Then is $\{\xi_n\}$ also a Cauchy sequence?

Answer 1

Yes. Use the Cauchy Schwarz inequality: $$ |\xi_m-\xi_n|=\langle x_n-x_m,\Phi\rangle\le |x_n-x_m|\cdot|\Phi| $$ So if $|x_n-x_m|<\epsilon$ for $n,m\ge N$, then $|\xi_m-\xi_n|\le \epsilon\cdot |\Phi|$ for $n,m\ge N$, so $\xi_n$ is Cauchy.