Operator Norm from Geometrical Point of View

Question

I'm trying to understand some Functional Analysis concepts from Geometrical point of view. So I know that Metric ($d(a,b)$) is the distance between elements, Norm ($||a||$) is the length of element.

So I'd like to know how Operator Norm ($||A||$, $A: X \to Y$) could be represented as Geometrical concept.

Answer 1

If $A:X\to Y$ is a linear bounded operator between the normed spaces $X,Y$, then the norm of $A$ is, loosely speaking, the factor by which the unit ball of $X$ gets inflated or deflated. More precisely, it is the smallest number $\alpha>0$ by which you need to dilate the unit ball of $Y$, so that $\alpha B_Y$ will contain the image of the unit ball of $X$ under the mapping $A$.

For example, a diagonal matrix $D=\{d_i\}_{i=1}^n$ in $\mathbb{R}^n$ maps the Euclidean unit ball to an ellipsoid, whose principal axis have lengths $2|d_i|$. The maximum of $|d_i|$ is of course the norm of this diagonal operator as a map between $\ell_2^n$ and itself. It is also the smallest number by which you need to multiply the unit ball of $\ell_2^n$ in order that the inflated ball will contain the ellipsoid $D(B_2^n)$, where $B_2^n$ is the unit ball of $\ell_2^n$.

Answer 2

Take the (closed) unit ball $B$. It is all the points of norm at most $1$.

Now, apply the operator $T$ to the unit ball, to get a convex set $T(B)$. The operator norm of $T$ is the radius of the smallest ball centered at $0$ that will contain $T(B)$. In finite dimensions, you think of $T$ turning the unit disk into some sort of ellipse-like region and the maximal distance from the origin of the boundary of this region is the operator norm.

For example, if $\|T\| = 0$, then that means that $T$ sends $B$ to the single point $0$, because the smallest closed ball containing $T(B)$ is in fact just the set $\{0\}$. We then have by linearity that $T$ is identically zero.

This is why the inequality $\|TS\| \leq \|T\|\|S\|$ is actually intuitive. Let $r$ be the radius of the smallest closed ball containing $TS(B)$ (i.e. the operator norm of $TS$). If we look at the smallest radius that contains $S(B)$, $s$, and apply $T$ to that radius ball and ask what is the smallest ball containing that set, get $\|T\|\|S\|$, but since this set clearly contains $TS(B)$, because instead of $T(S(B))$, we took $T(sB)$ and $S(B) \subset sB$, by definition of $s$.

Likewise, you can check geometrically the triangle inequality: $\|T + S\| \leq \|T\| + \|S\|$, because $(T + S)(B) \subset T(B) + S(B)$, because the first depends on adding only the points of the image that come from the same vector (i.e. $T(v) + S(v)$, but not $T(v) + S(w)$ if $v \neq w$) and the latter adds using all vectors in $B$ for arguments of $T$ and any vector in $B$ for an argument for $S$. Then we see that if $t$ is the smallest radius containing $T(B)$ and $s$ is the smallest radius contain $S(B)$, then the smallest radius containing $T(B) + S(B)$ is at most $t+s$, because $tB + sB = (t+s)B$, as sets.