# $(\cdot,\cdot)$ in Banach spaces?

I have been doing some research on fixed point theorems, and they have brought me around to papers from the 1960s in functional analysis in Banach spaces.

I think that today it is common practice to use either $\langle\cdot,\cdot\rangle$ or $(\cdot,\cdot)$ to mean an inner product in a Hilbert space. However, in these papers, they use the latter notation to mean something different: here, the first argument is a bounded conjugate linear functional from $V$, the second argument is in $V$, and $(f,x)=f(x)$.

Is there some reason that these two seemingly disparate concepts have the same notation? I do not know a lot of the language in linear algebra so I am a bit out of my league in terms of looking things up.

• It is actually still common to use $(f,x)=f(x)$ even today. This notation better illustrates the bilinear nature of the map $V^*\times V\to \mathcal{k}$. It is more symmetric and illustrates the duality between a banach space and its dual. – Hui Yu Jul 7 '13 at 1:41

If $V$ is a Banach space and $V^*$ its is dual, the duality operator $(f,v)=f(v)$ is a bilinear operator. Moreover if $V$ is a Hilbert space you can identify each $f\in V^*$ with some $v_f$ in $V$. In that case it happens that $(f,v) =\langle v_f,v\rangle$.