Banach space
Continuous and bounded linear functions and seminorms

Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a continuous function. So in particular, because the scalar field (which is R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} }) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.

If f: X → R {\displaystyle f:X\to \mathbb {R} } is a subadditive function (such as a norm, a sublinear function, or real linear functional), then f {\displaystyle f} is continuous at the origin if and only if f {\displaystyle f} is uniformly continuous on all of X {\displaystyle X}; and if in addition f (0) = 0 {\displaystyle f(0)=0} then f {\displaystyle f} is continuous if and only if its absolute value | f |: X → [ 0, ∞) {\displaystyle |f|:X\to [0,\infty)} is continuous, which happens if and only if { x ∈ X: | f (x) | < 1 } {\displaystyle \{x\in X:|f(x)|<1\}} is an open subset of X. {\displaystyle X.} And very importantly for applying the Hahn–Banach theorem, a linear functional f {\displaystyle f} is continuous if and only if this is true of its real part Re f {\displaystyle \operatorname {Re} f} and moreover, ‖ Re f ‖ = ‖ f ‖ {\displaystyle \|\operatorname {Re} f\|=\|f\|} and the real part Re f {\displaystyle \operatorname {Re} f} completely determines f, {\displaystyle f,} which is why the Hahn–Banach theorem is often stated only for real linear functionals.Also, a linear functional f {\displaystyle f} on X {\displaystyle X} is continuous if and only if the seminorm | f | {\displaystyle |f|} is continuous, which happens if and only if there exists a continuous seminorm p: X → R {\displaystyle p:X\to \mathbb {R} } such that | f | ≤ p {\displaystyle |f|\leq p}; this last statement involving the linear functional f {\displaystyle f} and seminorm p {\displaystyle p} is encountered in many versions of the Hahn–Banach theorem.
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