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Quotient Map is a Contraction Operator

Last updated Nov 1, 2022

# Statement

Let $(X, ||\cdot||)$ be a Banach Space

and let

Y \subset X

be a Closed Vector Subspace. Then the Quotient Map

J : X \to X / Y

defined as

J(x) := [x]

is a Bounded Linear Operator and

||J|| \leq 1

.

# Other Outlinks

Operator Norm