# About the Definition of Preserving Inner Products $(T \alpha|T \beta)=(\alpha|\beta)$

In the above definition, when it says $$(T \alpha|T \beta)=(\alpha|\beta)$$, the inner product $$(|)$$ acting on $$W$$ is the same as acting on $$V$$? Or not necessarily? The book uses the same notation, so I had this question.

• Not necessarily the same. – mechanodroid Feb 17 at 23:31
• What would that even mean were $V$ and $W$ not the same? – Alex Provost Feb 18 at 0:00

The inner product defined on $$V$$ may not be the same as the inner product defined on $$W$$, but given your $$\alpha$$ and $$\beta$$ in $$V$$, $$(\alpha|\beta)$$ is a just a number. $$T$$ preserves inner products if evaluating $$(T\alpha|T\beta)$$ gets you the same number as before.
It sounds like you could an example. Take $$V = \Bbb{R}^2$$, with $$(x \mid y)_V = x \cdot y$$, the dot product, and $$W = P_1(\Bbb{R})$$, the space of real polynomials of degree at most $$1$$, equipped with the inner product $$(p \mid q)_W = \int_{-1}^1 f(x)g(x) \; \mathrm{d}x.$$ Define $$T : V \to W$$ by $$T(a, b) = x \mapsto \frac{a}{\sqrt{2}} + \frac{\sqrt{3}b}{\sqrt{2}} x.$$ Then \begin{align*} (T(a, b) \mid T(c, d))_W &= \left(\left.\frac{a}{\sqrt{2}} + \frac{\sqrt{3}b}{\sqrt{2}} x \; \right| \; \frac{c}{\sqrt{2}} + \frac{\sqrt{3}d}{\sqrt{2}} x\right) \\ &= \int_{-1}^1 \left(\frac{a}{\sqrt{2}} + \frac{\sqrt{3}b}{\sqrt{2}} x\right)\left(\frac{c}{\sqrt{2}} + \frac{\sqrt{3}d}{\sqrt{2}} x\right) \; \mathrm{d}x \\ &= \left[\frac{ac}{2}x + \frac{1}{2}\left(\frac{\sqrt{3}b}{2} + \frac{\sqrt{3}d}{2}\right)x^2 + \frac{1}{2}bdx^3\right]_{x=-1}^{x=1} \\ &= ac + bd = ((a, b) \mid (c, d))_V \end{align*} That is, $$T$$ preserves inner products.
Of course, in the above example, I've really side-stepped your issue by differentiating between the two inner products $$(\cdot \mid \cdot)_V$$ and $$( \cdot \mid \cdot)_W$$. It is unfortunately common for different inner products on different spaces to have the same notation. You'd be expected to know which inner product to use from context.
For example, if I write (given the above example) $$((1, 1) \mid (0, 2))$$, I'm clearly talking about the dot product, and hence the number $$2$$. On the other hand, if I write $$(1 + x, x - 4)$$, I'm obviously using the inner product on $$W$$, which results in $$-\frac{22}{3}$$.