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On the linear span of $1,x$, define $g(a1+bx)=a+b\lambda$. Affine spectral mapping gives $|a+b\lambda |\le \Vert a1+bx\Vert$, making this well-defined with norm one and $g(1)=1$. A norm-preserving Hahn-Banach theorem extension to the whole C-star algebra remains unital; the norm characterization of a positive functional on a C-star algebra makes it a state on a C-star algebra attaining the spectral value.

For a bounded linear operator on a complex Banach space, either a spectral scalar has approximate unit eigenvectors or it is an eigenvalue of the Banach-space adjoint. Indeed outside the approximate point spectrum, $S=T-\lambda I$ is bounded below, hence injective with closed range. If the adjoint has no zero vector in its kernel other than zero, the Hahn-Banach theorem makes the range dense: any proper closed range would have a nonzero continuous functional annihilating it. Thus the range is all of the space and the lower bound gives a bounded inverse. Conversely invertibility rules out both alternatives. With complex-linear duals there is no conjugation of $\lambda$ in the displayed formula; Hilbert-adjoint conventions must be distinguished.

The sublinear functional $p(g)=\mu (\overline{g})$ has a linear supporting functional taking the value $p(f)$ at a specified continuous $f$, by the Hahn-Banach theorem. It is positive and normalized because $p$ has these values on constants. The Riesz-Markov-Kakutani representation theorem gives the probability $\nu$. Testing affine functions and their negatives shows that $\mu$ and $\nu$ have the same barycenter.

If the relative weak topology on the unit ball of a normed vector space is first countable at zero, choose a countable collection of finite families of continuous linear functionals defining a local base. For any $f\in E^{\prime }$ and $ϵ>0$, one family $F$ has common kernel $M$ on which $\Vert f{|}_M\Vert \le ϵ$. Extend that restriction to $h\in E^{\prime }$ with $\Vert h\Vert \le ϵ$ using the Hahn-Banach theorem. Then $f-h$ vanishes on $M$, so it belongs to $\mathrm{span}F$ by finite-dimensional factorization. Consequently the countable union of these families has norm-dense linear span in $E^{\prime }$. The converse is the metrization of the weak topology on a bounded set construction from a norm-dense sequence in $E^{\prime }$. In particular, separability of $E$ alone does not suffice: ${\ell }^1$ has nonseparable dual ${\ell }^{\infty }$.

A subset of a normed space is norm bounded if it is bounded under every continuous linear functional. Apply the Uniform boundedness principle on the complete dual space to the evaluation family $f\mapsto f(x)$. Hahn-Banach theorem identifies the evaluation norm with $\Vert x\Vert$, even when the original normed space is not complete. As a consequence, two norms with exactly the same continuous dual as sets must be equivalent: apply this criterion to either norm's unit ball in the other norm.

A bounded linear operator between Banach spaces is weakly compact exactly when its Banach-space adjoint is weakly compact. Equivalently, that adjoint is continuous from its domain's weak-star topology to its range's weak topology. For the reverse direction, the bidual condition applied to $T^{\ast }$ forces $T^{\ast \ast \ast }$ to annihilate every functional on $Y^{\ast \ast }$ vanishing on $J_{Y}Y$; the Hahn-Banach theorem then places $T^{\ast \ast }(X^{\ast \ast })$ inside $J_{Y}Y$.

For a weakly holomorphic Banach-valued function, choose a closed disk contained in its domain. Each continuous functional is bounded on the image of that disk by scalar continuity. The Uniform boundedness principle, applied to its evaluation functionals in the bidual, gives a common bound $M$ on the norms of the image vectors. Cauchy's formula on the disk's boundary then bounds the scalar difference between difference quotients at increments $h,k$ by a constant times $M\Vert \phi \Vert |h-k|$, uniformly for sufficiently small increments. Explicitly the difference kernel isThe Hahn-Banach theorem dual norm formula converts this into the same norm bound for the vector difference quotients. They are Cauchy, so completeness gives a norm derivative. This also explains why weak holomorphy suffices without a separately assumed norm-continuity hypothesis.

A countable local base at zero would give countably many finite sets of evaluation vectors. Any other evaluation must lie in the span of one of those finite sets: otherwise a bounded functional vanishing on that set but not on the evaluation vector, supplied by Hahn-Banach theorem, contradicts inclusion of neighbourhoods after scalar rescaling. Thus the predual has countable Hamel dimension, impossible for an infinite-dimensional Banach space by the Baire category theorem. This concerns the entire dual, unlike weak-star metrizability of the dual ball.