Let $u=f$, $v=g$ be continuous functions on $[0,1]$ where $f(x)=x^2+x$ and $g(x)=x+1$ with inner product $\langle f,g\rangle =\int_0^1f(x)g(x)dx$.
Find the distance between u and v.
We have only just started learning about the inner product and this question has thrown me off. My only thought is to find $\int_0^1(x^2+x)(x+1)dx = \int_0^1(x^3+2x^2+x)dx = \frac{17}{12} = $ distance between u and v?
Given some inner product $\langle , \rangle$ it induces a norm $\vert \cdot \vert$, canonically given by $\vert x \vert = \sqrt{\langle x, x \rangle}$. Then the distance function $d(x,y)$ becomes $\vert y-x \vert$.
In this case you're calculating $\vert v-u \vert = \vert u-v|=\sqrt{\langle x^2-1, x^2-1 \rangle}$ and you seem capable of finishing from here. I found this question confusing because they reuse the letters $f$ and $g$ when defining the inner product even though they used them earlier to define $u$ and $v$. It's misleading and unnecessary.