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.