Question: A linear function f: X Y is bounded if X has finite dimension. Rewriting (4), we have that a linear function f is bounded if

A linear function f: X †’ Y is bounded if X has finite dimension.
Rewriting (4), we have that a linear function f is bounded if there exists a constant M

For every x ˆŠ X
The smallest constant M satisfying (5) is called the norm of f. It is given by

Clearly,

f(x) llf'll = sup Ilf(x)11

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