Definition:Compact Linear Transformation/Normed Vector Space/Definition 2

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.

Let $T : X \to Y$ be a linear transformation.


We say that $T$ is a compact linear transformation if and only if:

for each bounded sequence $\sequence {x_n}_{n \mathop \in \N}$ in $X$:
the sequence $\sequence {T x_n}_{n \mathop \in \N}$ has a subsequence convergent in $\struct {Y, \norm \cdot_Y}$.


Also see


Sources