Riesz Representation Theorem and Uniqueness

Question

I'm working through the proof of the Reisz Representation Theorem in Axler's Linear Algebra Done Right and am confused about the uniqueness portion of the proof.

Axler claims:

"Suppose $V$ is finite-dimensional and $\phi $ is a linear functional on V. Then there is a unique vector $u \in V$ such $$ \phi(v) = \langle v,u \rangle$$ for every $v \in V$."

The text then proceeds to prove existence (which I understand, so am not going to elaborate here) and then uniqueness of $ u$.

To prove uniqueness, the text claims the following:

Now we prove that only one vector $ u \in V $ has the desired behavior.

Suppose $ u_1, u_2 \in V $ are such that $$ \phi(v) = \langle v,u_1 \rangle = \langle v,u_2 \rangle $$ for every $v \in V$. Then $$ 0 = \langle v,u_1 \rangle - \langle v,u_2 \rangle = \langle v,u_1-u_2 \rangle $$ for every $ v \in V$. Taking $ v = u_1 - u_2 $ shows that $ u_1 - u_2 = 0 $. In other words, $u_1 = u_2$, completing the proof.

Some questions about this:

By "taking $ v = u_1 - u_2 $", I assume the text means that we set $ v = u_1 - u_2 $. However, why does $ 0 = \langle u_1 - u_2 , u_1 - u_2 \rangle \implies u_1 - u_2 = 0 $?

Isn't $\langle u_1 - u_2 , u_1 - u_2 \rangle = (u_1 - u_2)^T (u_1 - u_2) $? As a result, isn't $\langle u_1 - u_2 , u_1 - u_2 \rangle = 0 $ iff $V$ is a vector space over $\mathbb{R^n}$? (The text assumes that $V$ is a vector space over $\mathbb{R^n}$ or $\mathbb{C^n}$.)

(I realize that this might be more of a question about whether $AA^T = 0 \implies A = 0$, and I've already looked at a similar question about that. However, I've included details of the proof, just in case my interpretation of what it was saying was wrong. I've also looked at other questions that also discuss this line of the proof, but am still confused about the calculation above.)

Thanks!

Answer 1

A scalar product $\langle\cdot,\cdot\rangle$ induces on a vector space the norm $$ \|v\|=\sqrt{\langle v,v\rangle}. $$ One of the properties of the norm is that $$ \|v\|=0\Leftrightarrow v=0. $$ Then, specialising this property to the proof of the theorem, we have $$ \langle u_1-u_2, u_1-u_2\rangle=0\Leftrightarrow \|u_1-u_2\|=0 \Leftrightarrow u_1=u_2. $$ This is a trick used very often in Hilbert spaces.

Answer 2

If $\phi(v)=\langle v,u_1\rangle=\langle v,u_2\rangle$, then $\langle v,u_1-u_2\rangle=0$ for any choice of $v\in V$. So, cleverly choose $v=u_1-u_2$. Then $$ 0=\langle u_1-u_2,u_1-u_2\rangle.$$ One of the conditions in the definition of an inner product is that $\langle w,w\rangle=0$ if and only $w=0$. So, $u_1-u_2=0$ and $u_1=u_2$. As a concrete example, let's examine the standard inner product $\langle x,y\rangle=x\cdot y$ on $\mathbb{R}^3$. If $x_i$ denote the components of $x$ and $y_i$ those of $y$, then $$ x\cdot y=\sum_{i=1}^3 x_iy_i=x^Ty.$$ Note that $$ \langle x,x\rangle=x\cdot x=\sum_{i=1}^3x_i^2\ge 0$$ and equals zero exactly when $x_1=x_2=x_3=0$.