Explaining maths

In this part we'll have an initial look at some of the connections between real inner product spaces and complex inner product spaces. In a previous post, I discussed the fact that every vector space over $\C$ can be regarded as a vector space over $\R$ with respect to the same addition, and the restricted scalar multiplication obtained by using only real scalars (where we regard $\R$ as a subset of $\C$, as usual). If we have a finite-dimensional vector space over $\C$, then the dimension doubles when we regard it as a vector space over $\R$. In particular, $\C$ is a $1$-dimensional complex vector space, but is a $2$-dimensional real vector space. This is no surprise, since we often identify $\C$ with $\R^2$ (and may well define $\C$ that way as a set). Similarly, every normed space over $\C$ can also be regarded as a normed apace over $\R$, with the same addition and norm, and restricting attention to multiplication by only real scalars. Now, however, we are talking about inner p...

I'm currently thinking about the Cauchy-Schwarz inequality for multiple reasons, including the following: I'm currently teaching a Level 4 module on Linear Analysis which includes material on inner product spaces and Hilbert spaces; I'm also still the Teaching Support Officer in our school, and one of the first-year students asked me why the dot product formula in terms of components gives the same answer as the formula in terms of norms and angles, as in $$\va \cdot \vb = ||\va||\cdot||\vb|| \cos \theta,$$ where $\theta$ is the angle between $\va$ and $\vb.$ For the first-year student, I drew a picture to illustrate the special case in $\R^2$  where $\va$ has the form $(||\va||,0)$ and $\vb$ is written in the form $(||\vb|| \cos \theta,||\vb||\sin \theta).$ I then noted that the two formulae clearly agreed here, and so, as long as you know that the dot product formulae are invariant under rotations about the origin , the general fact follows.  Now I'm ...