3

In most literature on functional analysis the notation for an inner product on an inner product space $H$ is usually denoted $ \langle\cdot,\cdot\rangle:H \times H \mapsto \mathbb{K} $. However, I have noticed that some authors perfers to write it as $ (\cdot | \cdot) $. Searching on the internet this leads me to believe that this has something to do with notations in physic, namely the bra-ket which are frequently applied in quantum mechanics.

However what seems interesting to me more is that, even if $H$ is just a Banach space without an inner product, sometimes a functionl on the dual space $H^{\ast}$ of $H$ is denoted as follows: $$f:H^{\ast} \longrightarrow \mathbb{K} : x \longmapsto f(x) \Longleftrightarrow \langle f,x\rangle: H^{\ast} \times H \longrightarrow \mathbb{K} $$ which actually seems super useful because it has this analogy with the Cauchy Schwarz inequality： $$| \langle f,x\rangle | \leq \| f\|_{\text{op}} \|x\|_{H} $$ since $f$ is bounded. Also if $T^{\ast}$ is the dual operator of $T:H \rightarrow H$ then one also has $$\langle T^{\ast}f,x\rangle = \langle f, Tx\rangle $$ which in some sense resembles the notion of an adjoint.

So my question is: what are these notations called and where they come from? In particular do they actually have anything to do with the notion of inner product?

Thanks in advance!

Martin Argerami
  • 205,756
Meagain
  • 693
  • A difference between the $\left\langle \cdot,\cdot\right\rangle$ notation is that it is bilinear in Banach spaces, but sesquilinear in (complex) Hilbert spaces (i.e. linear in one component, antilinear in the other). – Lorenzo Q May 25 '18 at 12:56

1 Answers1

1

Riesz representation theorem tells us for every $f \in H^*$ there exists an $y \in H$ such that $f(x) = \langle y,x \rangle$ for every $x \in H$, so this notation makes sense for Hilbert spaces.

eddie
  • 640
  • 4
  • 10