2.3 Operator norms and continuity

Operator norms are crucial tools in functional analysis, measuring the "size" of linear operators between normed spaces. They possess key properties like positivity, homogeneity, and subadditivity, which make them invaluable for analyzing operator behavior.

These norms connect to broader concepts like continuity and boundedness of linear operators. They also play a role in convergence of operator sequences and series, providing a framework for understanding limits in operator spaces.

Operator Norms

Properties of operator norms

• Positivity ensures operator norms are non-negative and zero only for the zero operator
• Homogeneity scales the operator norm by the absolute value of a scalar multiple
• Subadditivity (Triangle inequality) bounds the operator norm of a sum by the sum of operator norms
• Submultiplicativity bounds the operator norm of a composition by the product of operator norms
• Equivalence of operator norms means any two operator norms are within constant factors of each other (on the same space)

Continuity of bounded linear operators

• Theorem establishes the equivalence between continuity and boundedness for linear operators between normed spaces
  • Bounded implies Lipschitz continuous with the operator norm as the Lipschitz constant
  • Continuous implies bounded by considering the continuity condition at the origin with $\varepsilon = 1$

Calculation of operator norms

• $\ell^1$ operator $(Tx)_n = \frac{x_n}{n}$ has operator norm at most 1 by comparing $\ell^1$ norms of $Tx$ and $x$
• $C[0, 1]$ integral operator $(Tf)(x) = \int_0^x f(t) dt$ has operator norm at most 1 by bounding the supremum of $|Tf|$ by the supremum of $|f|$

Operator norms and sequence convergence

• Theorem relates convergence in operator norm to pointwise convergence and boundedness of the limit operator
  • Convergence in operator norm implies pointwise convergence and boundedness of the limit
• Corollary provides a sufficient condition for a series of operators to converge in operator norm
  • Summable operator norms imply convergence to a bounded operator by completeness of the operator space