# Metric Slope and Subdifferential Calculus in Open image in new window(X)

Part of the Lectures in Mathematics ETH Zürich book series (LM)

## Abstract

As we have seen in Section 1.4, in the classical theory of subdifferential calculus for proper, lower semicontinuous functionals φ : X → (−∞,+∞] defined in a Hilbert space X, the Fréchet Subdifferential ∂φ : X → 2 X of φ is a multivalued operator defined as
$$\xi \in \partial \varphi \left( \upsilon \right) \Leftrightarrow \upsilon \in D\left( \varphi \right),\mathop {\lim \inf }\limits_{w \to \upsilon } \frac{{\varphi \left( w \right) - \varphi \left( \upsilon \right) - \left\langle {\xi ,w - \upsilon } \right\rangle }} {{\left| {w - \upsilon } \right|}} \geqslant 0,$$
(10.0.1)
which we will also write in the equivalent form for vD(φ)
$$\xi \in \partial \varphi \left( \upsilon \right) \Leftrightarrow \varphi \left( w \right) \geqslant \varphi \left( \upsilon \right) + \left\langle {\xi ,w - \upsilon } \right\rangle + o\left( {\left| {w - \upsilon } \right|} \right)asw \to \upsilon .$$
(10.0.2)
As usual in multivalued analysis, the proper domain D(∂φ) ⊂ D(φ) is defined as the set of all vX such that ∂φ(v) ≠ φ; we will use this convention for all the multivalued operators we will introduce.

## Keywords

Variational Inequality Lower Semicontinuity Doubling Condition Multivalued Operator Minimal Selection
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