We prove that for Hilbert space operators
X
X
and
Y
Y
, it follows that
\[
lim
t
→
0
+
|
|
X
+
t
Y
|
|
−
|
|
X
|
|
t
=
1
|
|
X
|
|
inf
ε
>
0
sup
φ
∈
H
ε
,
|
|
φ
|
|
=
1
Re
⟩
Y
φ
,
X
φ
⟩
,
\lim _{t\to 0^+}\frac {||X+tY||-||X||}t=\frac 1{||X||} \inf _{\varepsilon >0}\sup _{\varphi \in H_\varepsilon ,||\varphi ||=1} \operatorname {Re}\left >Y\varphi ,X\varphi \right >,
\]
where
H
ε
=
E
X
∗
X
(
(
|
|
X
|
|
−
ε
)
2
,
|
|
X
|
|
2
)
H_\varepsilon =E_{X^*X}((||X||-\varepsilon )^2,||X||^2)
. Using the concept of
φ
\varphi
-Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in
B
(
H
)
B(H)
, and to give an easy proof of the characterization of smooth points in
B
(
H
)
B(H)
.