Inner product identity not understood

Question

Reading Axler Linear Algebra Done Right, page 205, about determining the representing vector in Riesz representation of a linear functional, We're given this (rephrased):

• $V$ is an inner product space with an orthonormal basis $e_1, \dotsb, e_m$
• $\varphi$ is a linear functional of $V$, $v \in V$

$$ \begin {align} \varphi(v) &= \left\langle v, e_{1} \right\rangle \varphi(e_{1}) + \dotsb + \left\langle v, e_{m} \right\rangle \varphi(e_{m}) \\ &= \left\langle v, \overline {\varphi(e_{1})} e_{1} + \dotsb + \overline {\varphi(e_{m})} e_{m} \right\rangle \end {align} $$

and with that $\varphi(v) = \left \langle v, u \right \rangle$ with $u = \overline {\varphi(e_{1})} e_{1} + \dotsb + \overline {\varphi(e_{m})} e_{m}$

I can't retrieve the rule allowing to replace the sum of inner products by the single inner product. How should that be understood?

Answer 1

I'm posting this answer in case someone could be as ignorant as I'm. As suggested by @dvdgrgrtt in a comment, I reviewed the properties of the inner product presented in the book:

• Additivity in the first slot: $\langle u+v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$

• homogeneity in first slot: $\langle \lambda u, v \rangle = \lambda \langle u, v \rangle$

None can be applied directly here. Instead we need the complements: Additivity in the second slot and antihomogeneity in the second slot. They are not explained in the book, but are here.

• $\langle u, v+w \rangle = \langle u,v \rangle + \langle u, w \rangle$

• $\langle u, \lambda v \rangle = \overline \lambda \langle u, v \rangle$

---

$$ \begin {align} \varphi(v) &= \left\langle v, e_{1} \right\rangle \varphi(e_{1}) + \dotsb + \left\langle v, e_{m} \right\rangle \varphi(e_{m}) \\ &= \color {mediumblue}{ \left\langle v, \overline {\varphi(e_{1})} e_{1} \right\rangle + \dotsb + \left\langle v, \overline {\varphi(e_{m})} e_{m} \right\rangle }\\ &= \left\langle v, \overline {\varphi(e_{1})} e_{1} + \dotsb + \overline {\varphi(e_{m})} e_{m} \right\rangle \end {align} $$

Answer 2

On page 183 of the Axler's Linear Algebra Done Right a definition of an inner product is given:

Definition: An inner product on [a vector space] $V$ is a function that takes each ordered pair $(u,v)$ of element of $V$ to a number $\langle u,v\rangle \in F$ and has the following properties:

• positivity: $\langle v,v\rangle \ge 0$ for all $v\in V$;
• definiteness: $\langle v,v\rangle = 0$ if and onlhy if $v = 0$;
• additivity in the first slot: $\langle u+v,w\rangle = \langle u,w\rangle + \langle v,w\rangle$ for all $u,v,w\in V$;
• homogeneity in the first slot: $\langle \lambda u v\rangle = \lambda \langle u,v\rangle$ for all $\lambda \in F$ and all $u,v\in V$; and
• conjugage symmetry: $\langle u,v \rangle = \overline{\langle u,v\rangle}$ for all $u,v\in V$.

There are two key ideas here:

• First, an inner product is linear in the second slot, as well: if $u,v,w \in V$, then

$$\begin{align} \langle u, v+w \rangle &= \overline{\langle v+w, \rangle} && \text{(conjugate symmetry)} \\ &= \overline{\langle v,u\rangle + \langle w, u \rangle} &&\text{(additivity in the first slot)} \\ &= \overline{\langle v,u\rangle} + \overline{\langle w,u\rangle} \\ &= \langle u, v\rangle + \langle u, w\rangle. \end{align}$$

• Second, an inner product is conjugate linear (or anti-homogeneous) in the second slot: if $\lambda \in F$ and $u,v\in V$, then \begin{align} \lambda \langle u,v\rangle &= \lambda \overline{\langle v,u\rangle} && \text{(conjugate symmetry)} \\ &= \overline{\overline{\lambda} \langle v,u \rangle} \\ &= \overline{\langle \overline{\lambda}v,u\rangle} && \text{(homogeneity in the first slot)} \\ &= \langle u, \overline{\lambda}v \rangle. && \text{(conjugage symmetry)} \end{align}

Note that both of these properties are discussed on page 185, and proofs are given. These two properties can be applied to get the desired result: $$ \begin {align} \varphi(v) &= \left\langle v, e_{1} \right\rangle \varphi(e_{1}) + \dotsb + \left\langle v, e_{m} \right\rangle \varphi(e_{m}) \\ &= \left\langle v, \overline{\varphi(e_1)}e_1 \right\rangle + \dotsb + \left\langle v, \overline{\varphi(e_n)}e_n \right\rangle && \text{(anti-homogeneity)} \\ &= \left\langle v, \overline {\varphi(e_{1})} e_{1} + \dotsb + \overline {\varphi(e_{m})} e_{m} \right\rangle. && \text{(linearity in the second term)} \end {align} $$