12.4 Norms of Linear application[2CM]

In the following \((X,\| \| _ X)\)and \((Y,\| \| _ Y)\) will be normed spaces; \(A:X→ Y\) is a linear application; we define the induced norm as

E338

[11B]Show that \(\| A\| _{X,Y}{\lt}∞\) if and only if \(A\) is continuous.

E338

[11C]Note that if \(X\) has finite dimension then every linear application is continuous, and

\[ \| A\| _{X,Y}= \max _{x∈ X~ ,~ \| x\| _ X≤ 1} \| Ax\| _ Y~ ~ . \]

E338

[11D]Let \({\mathcal L}(X,Y)\) be the space of all continuous linear applications. Show that \(\| ⋅ \| _{X,Y}\) is a norm in \({\mathcal L}(X,Y)\).

E338

[11F] Let \((Z,\| \| _ Z)\) be an additional normed space, and \(B:Y→ Z\) a linear application. We similarly define

\[ \| B\| _{Y,Z}{\stackrel{.}{=}}\sup _{y∈ Y~ ,~ \| y\| _ Y≤ 1} \| B y \| _ Z\quad ; \]

show that

\[ \| A B\| _{X,Z} ≤ \| A\| _{X,Y} \| B\| _{Y,Z}~ ~ . \]