# Definition:Space of Bounded Linear Transformations

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## Definition

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.

Then the **space of bounded linear transformations from $H$ to $K$**, $\map B {X, Y}$, is the set of all bounded linear transformations:

- $\map B {X, Y} := \set {A: X \to Y: A \text{ linear}, \text { there exists } M > 0 \text { such that } \norm {A x}_Y \le M \norm x_X \text { for all } x \in X}$

endowed with pointwise addition and ($\GF$)-scalar multiplication.

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By Space of Bounded Linear Transformations forms Vector Space, $\map B {X, Y}$ is a vector space over $\Bbb F$.

Furthermore, let $\norm {\, \cdot \,}_{\map B {X, Y} }$ denote the norm on the space of bounded linear transformations.

Then, by Norm on Space of Bounded Linear Transformations is Norm $\norm {\, \cdot \,}_{\map B {X, Y} }$ is indeed a norm on $\map B {X, Y}$.

From Space of Bounded Linear Transformations is Banach Space, $\struct {\map B {X, Y}, \norm {\, \cdot \,}_{\map B {X, Y} } }$ is then a Banach space if $Y$ is a Banach space.

### Space of Bounded Linear Operators

When $X = Y$, one denotes $\map B X$ for $\map B {X, Y}$.

In line with the definition of linear operator, $\map B X$ is called the **space of bounded linear operators on $X$**.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next) $\S \text {II}.1$