The norm $\| \cdot \|$ associated with the inner product $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ is defined for all functions $f \in E$ by $$\| f \| = \left( \int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t \right) ^ { 1 / 2 }$$ Show that $\lim _ { x \rightarrow 0 } \left\| k _ { x } \right\| = 0$. We recall that, for all $x > 0 , k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$.
The norm $\| \cdot \|$ associated with the inner product $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ is defined for all functions $f \in E$ by
$$\| f \| = \left( \int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t \right) ^ { 1 / 2 }$$
Show that $\lim _ { x \rightarrow 0 } \left\| k _ { x } \right\| = 0$. We recall that, for all $x > 0 , k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$.