# Help explain the geometry being described in this paragraph

In A First Course in Optimization Theory by Sundaram I read on page 51:

Given a quadratic form $A$ and any $t\in\mathbb R$, we have $(tx)'A(tx)=t^2x'Ax$, so the quadratic form has the same sign along lines through the origin. Thus, in particular, $A$ is positive definite (resp. negative definite) if and only if it satisfies $x'Ax>0$ (resp. $x'Ax<0$) for all $x$ in the unit sphere $\mathcal C=\{u\in\mathbb R^n\mid\lVert u\rVert=1\}$ We will use this observation to show that if $A$ is a positive definite (or negative definite) $n\times n$ matrix, so is any other quadratic form $B$ which is sufficiently close to $A$:

I don't understand the the part where it says "for all $x$ in the unit sphere $\mathcal C=\{u\in\mathbb R^n\mid\lVert u\rVert=1\}$".

Shouldn't the $u$ in the set $\mathcal C$ be in $\mathbb{R}^3$ since $\mathcal C$ is supposed to describe a unit sphere?

Would be great if there is a diagram that depicts the thing being described in this paragraph.

• I'm not sure I understand the question, but are you aware that a 'sphere' is simply a set that looks like $\{v\in \mathbb R^n\colon \Vert v-v_0\Vert =r\}$, independent of what $n$ is? For instance with $v_0=0, n=1$ and $r=1$ the sphere is simply $\{-1,1\}$. – Git Gud Oct 27 '14 at 11:39
• @GitGud No, I was not aware of it. I always thought a 'sphere' refers to that round object in 3D. Your comment makes the meaning of the paragraph clear to me. Thank you. – mauna Oct 27 '14 at 11:48
• See this wikipedia entry. – Git Gud Oct 27 '14 at 11:49

“Sphere” is a concept which applies to any dimension. It is the set of points which have the same distance (radius) from a given point (center). For the unit sphere, the radius is $1$ and the center is the origin. So the set of all vectors with unit norm (i.e. length one) forms the unit sphere, in any dimension.