User:Caliburn/s/fa/Definition:Space of Bounded Linear Transformations

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Definition

Normed Vector Space

Let $\struct {V, \norm \cdot_V}$ and $\struct {U, \norm \cdot_U}$ be normed vector spaces.


Then the space of bounded linear transformations from $V$ to $U$, $\map B {V, U}$, is defined by:

$\map B {V, U} = \set {A : V \to U \mid A \text { is a bounded linear transformation} }$


Inner Product Space

Let $\struct {V, \innerprod \cdot \cdot_V}$ and $\struct {U, \innerprod \cdot \cdot_U}$ be inner product spaces.


Then the space of bounded linear transformations from $V$ to $U$, $\map B {V, U}$, is defined by:

$\map B {V, U} = \set {A : V \to U \mid A \text { is a bounded linear transformation} }$


Also see