Definition:Bounded Linear Transformation/Inner Product Space
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Definition
Let $\struct {V, \innerprod \cdot \cdot_V}$ and $\struct {U, \innerprod \cdot \cdot_U}$ be inner product spaces.
Let $\norm \cdot_V$ and $\norm \cdot_U$ be the inner product norms of $V$ and $U$ respectively.
Let $A : V \to U$ be a linear transformation.
We say that $A$ is a bounded linear transformation if and only if:
- there exists $c > 0$ such that $\norm {A v}_U \le c \norm v_V$ for each $v \in V$.
Also see
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\S II.1$