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
- Space of Bounded Linear Transformations forms Vector Space shows that $\map B {V, U}$ forms a vector space with pointwise addition and pointwise scalar multiplication.
- Norm on Space of Bounded Linear Transformations is Norm shows that $\map B {V, U}$ forms a normed vector space.
- Space of Bounded Linear Transformations is Banach Space shows that $\map B {V, U}$ in fact forms a Banach space.