Definition:Compact Linear Transformation/Inner Product Space/Definition 2

Definition

Let $\struct {X, \innerprod \cdot \cdot_X}$ and $\struct {Y, \innerprod \cdot \cdot_Y}$ be inner product spaces.

Let $\norm \cdot_X$ and $\norm \cdot_Y$ be the inner product norms of $\struct {X, \innerprod \cdot \cdot_X}$ and $\struct {Y, \innerprod \cdot \cdot_Y}$ respectively.

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

• Equivalence of Definitions of Compact Linear Transformation on Inner Product Space

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