What is a $\langle , \rangle_{H}$? And how do you prove that it is an inner product? We were given this problem:
For the inner product space $V$ and its subspace $H$, let $Q = \{E(v)\,|\,v\in V\}$ be the vector space.
Define addition in $Q$ by: for $v, w\in V, E(v)\oplus E(w) = E(v + w)$.
Define scalar multiplication in Q by: for $v \in V$ and $\rho\in \mathbb{R}, \rho E(v) = E(\rho v)$.
For $v\in V$, define $E(v) = \{v+h\,|\,h\in H\}$.
Define for $u,v\in V$, and $u = u_1 + u_2, v = v_1 + v_2$ where $u_1, v_1 \in H$ and $u_2, v_2\in H^{\bot}$,
\begin{align*}
\langle E(u), E(v)\rangle_H = \langle u_2, v_2\rangle
\end{align*}
Prove or disprove: $\langle \, ,\,\rangle_H$ is an inner product on $Q$.
What is $\langle \,,\,\rangle_H$? Anyway, here are my initial thoughts in solving the problem:
To prove $\langle , \rangle_H$ is an inner product on $Q$, we have to prove the following:

*

*For all $E(a), E(b), E(y)\in V$, $\langle E(a), E(b) + E(y)\rangle_H = \langle E(a), E(b)\rangle + \langle E(a), E(y)\rangle_H$.
\begin{align*}
            \langle E(a), E(b) + E(y)\rangle_H = E(a)_H \cdot E(b + y)_H &= \begin{pmatrix}
            a_1\\
            a_2\\
            \vdots\\
            a_n
            \end{pmatrix}\cdot \begin{pmatrix}
            b_1 + y_1\\
            b_2 + y_2\\
            \vdots\\
            b_n + y_n
            \end{pmatrix}\\
            &= \sum_{i=1}^{n} a_i(b_i+y_i)\\
            &= \sum_{i=1}^{n} a_ib_i + \sum_{i=1}^{n} a_iy_i\\
            &= \begin{pmatrix}
            a_1\\
            a_2\\
            \vdots\\
            a_n
            \end{pmatrix} \cdot \begin{pmatrix}
            b_1\\
            b_2\\
            \vdots\\
            b_n
            \end{pmatrix} + 
            \begin{pmatrix}
            a_1\\
            a_2\\
            \vdots\\
            a_n
            \end{pmatrix} \cdot
            \begin{pmatrix}
            y_1\\
            y_2\\
            \vdots\\
            y_n
            \end{pmatrix}\\
            &= E(a)_H\cdot E(b)_H + E(a)_H \cdot E(y)_H\\
            &= \langle E(a), E(b)\rangle + \langle E(a), E(y)\rangle_H
        \end{align*}

*For all vectors $a, b\in V$, $\langle a, b\rangle = \langle b, a\rangle$.
\begin{align*}
            \langle a, b\rangle = E(a)_H\cdot E(b)_H &= \begin{pmatrix}
            a_1\\
            a_2\\
            \vdots\\
            a_n
            \end{pmatrix}\cdot\begin{pmatrix}
            b_1\\
            b_2\\
            \vdots\\
            b_n
            \end{pmatrix}\\
            &= \sum_{i=1}^{n} a_ib_i\\
            &= \sum_{i=1}^{n} b_ia_i\\
            &= \begin{pmatrix}
            b_1\\
            b_2\\
            \vdots\\
            b_n
            \end{pmatrix}\cdot \begin{pmatrix}
            a_1\\
            a_2\\
            \vdots\\
            a_n
            \end{pmatrix}\\
            &= E(b)_H\cdot E(a)_H\\
            &= \langle b, a\rangle
        \end{align*}

*For all vectors $a, b$ and for all real numbers $r \in \mathbb{R}, \langle ra,b\rangle = r\langle a,b\rangle $.
\begin{align*}
            \langle ra, b\rangle = E(ra)_H\cdot E(b)_H &= \begin{pmatrix}
            ra_1\\
            ra_2\\
            \vdots\\
            ra_n
            \end{pmatrix}\cdot \begin{pmatrix}
            b_1\\
            b_2\\
            \vdots\\
            b_n
            \end{pmatrix}\\
            &= \sum_{i=1}^{n} (ra_i)b_i\\
            &= r\sum_{i=1}^{n} a_ib_i\\
            &= r\begin{pmatrix}
            a_1\\
            a_2\\
            \vdots\\
            a_n
            \end{pmatrix}\cdot \begin{pmatrix}
            b_1\\
            b_2\\
            \vdots\\
            b_n
            \end{pmatrix}\\
            &= r(E(a)_H\cdot E(b)_H)\\
            &= r\langle a,b\rangle
        \end{align*}

*For all vectors $a\in V, \langle a,a \rangle \geq 0$ and $\langle a,a \rangle = 0$ iff $a = 0$.
\begin{align*}
            \langle a, a\rangle = E(a)_H \cdot E(a)_H &= \begin{pmatrix}
            a_1\\
            a_2\\
            \vdots\\
            a_n
            \end{pmatrix}\cdot \begin{pmatrix}
            a_1\\
            a_2\\
            \vdots\\
            a_n
            \end{pmatrix}\\
            &= \sum_{i=1}^{n} {a_i}^2 \geq 0
        \end{align*}
But then again, I have no idea what $\langle\,,\,\rangle_H$ is.

 A: To begin with, $E(v)$ is a subset of $V$. Let us call such a set an $E$-set.
Next, $\langle \cdot, \cdot \rangle_H$ is a function that takes two $E$-sets and returns a number; given $E(u)$ and $E(v)$ it returns $\langle u_2, v_2 \rangle,$ where $u_2,v_2$ are the $H^\bot$ components of $u,v$ respectively.
Then, if you can show that $\langle \cdot, \cdot \rangle_H$ satisfies the criteria for being an inner product, then you have an inner product on the set of $E$-sets. You can then say things like "this $E$-set is orthogonal to this other $E$-set."


To prove $\langle , \rangle_H$ is an inner product on $Q$, we have to prove the following:

*

*For all vectors $a, b,$ and $y\in V$, $\langle a, (b + y)\rangle = \langle a,b\rangle + \langle a + y\rangle$.


Not for all $a,b,y\in V$ but for all $E$-sets $a,b,y,$ or
$$
\langle E(u), E(v)\oplus E(w) \rangle_H = \langle E(u), E(v) \rangle + \langle E(u), E(w) \rangle_H
$$
for $u,v,w\in V.$
Proof
$$
\langle E(u), E(v) \oplus E(w) \rangle_H
= \langle E(u), E(v+w) \rangle_H
= \langle u_2, (v+w)_2 \rangle
= \langle u_2, v_2+w_2 \rangle
\\
= \langle u_2, v_2 \rangle + \langle u_2, w_2 \rangle
= \langle E(u), E(v) \rangle_H + \langle E(u), E(w) \rangle_H.
$$
Note that you need to show first that $(v+w)_2=v_2+w_2.$ Everything else follows from definition of $\langle \cdot, \cdot \rangle_H$ and from $\langle \cdot, \cdot \rangle$ being an inner product on $V.$
