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The sum consists of measurable functions $f=g+h$ with $g\in L^{p_0}$ and $h\in L^{p_1}$, with norm ${\inf }_{f=g+h}(\Vert g{\Vert }_{p_0}+\Vert h{\Vert }_{p_1})$. It is a Banach space. One construction identifies it with the quotient of $L^{p_0}{\oplus }_1{L}^{p_1}$ by the closed kernel of the addition map. Bounded endpoint operators are therefore bounded on the corresponding sum spaces.

For a countable minimal total family in a normed space in a Banach space, let $(x_n,f_n)$ be its unique biorthogonal system and let $P_{n}x={\sum }_{j\le n}{f}_j(x)x_j$. Uniform boundedness of these projections implies $P_{n}x\to x$ for every $x$: approximate $x$ by a finite linear combination $y$, eventually fixed by $P_n$, and bound the error by $(1+{\sup }_n\Vert P_n\Vert )\Vert x-y\Vert$. Conversely, if every vector has a norm-convergent expansion in this family, applying each continuous $f_j$ shows that its coefficients must be $f_j(x)$. Thus $P_{n}x\to x$, so the projections are pointwise bounded. The Uniform boundedness principle gives their uniform operator-norm bound. This proves the criterion and uniqueness of all expansions.

Let $K$ be a convex set in a real Banach space, and let $A:K\to X^{\ast }$ be a strongly monotone operator. If $u,w\in K$ both satisfy the displayed variational inequality, testing with $v=w$ and $v=u$ respectively and adding yields $⟨Au-Aw,u-w⟩\le 0$. Strong monotonicity gives $m\Vert u-w{\Vert }^2\le 0$, hence $u=w$. This proves uniqueness whenever a solution exists.

A Banach space is reflexive if and only if its closed unit ball is compact in the weak topology. One direction applies the Banach-Alaoglu theorem to the unit ball in the bidual; the converse follows because Goldstine theorem makes the canonical image of the unit ball weak-star dense, while weak compactness makes that image weak-star closed.

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 fixed-endpoint integral on $W^{1,\infty }(a,b)$, an admissible function $u$ is a weak local minimizer if $I(v)\ge I(u)$ whenever the Sobolev norm $\Vert v-u{\Vert }_{W^{1,\infty }}$ is sufficiently small. Both function and derivative must be close. “Weak” here labels the variational neighborhood convention, not the weak topology of a Banach space.

A map from an open subset of $\mathbb{C}$ into a complex Banach space is weakly holomorphic when every composition with a bounded complex-linear functional is a scalar holomorphic function. Norm holomorphy requires its difference quotients to converge in the Banach norm. Weak holomorphy is initially a condition on scalar observations; the weak holomorphy implies norm holomorphy argument recovers an actual vector derivative. A closed target subspace containing values on a sequence with an interior accumulation point contains every value on the connected domain: annihilate the subspace, apply the scalar identity theorem, and use Hahn-Banach separation.

If $T=S^{\ast }$ for a bounded operator on a separable Banach space, and $C$ is a nonempty weak-star closed convex subset of the dual unit ball with $T(C)\subseteq C$, then $T$ has a fixed point in $C$. The Cesàro averages of the orbit remain in $C$, and $T$ minus the identity sends those averages to vectors of norm at most $2/n$. Weak-star compactness supplies a cluster point, and the adjoint action is weak-star continuous. The corresponding assertion with only weak closedness is false for the right shift on the probability simplex in ${\ell }^1$, the dual of $c_0$.

The continuous dual of a separable Banach space is separable in its weak-star topology, even when it is not norm separable. Banach-Alaoglu theorem and weak-star metrizability of the dual ball make the dual unit ball a separable compact metric space. The countable union of positive integer multiples of a countable dense subset of that ball is dense in the whole dual. This does not imply global weak-star metrizability.

The same Banach space can acquire incomparable weak-star topologies from different isometric preduals. Pair ${\ell }^1$ with $c_0$ by $\sum {\alpha }_n{\beta }_n$ and with the space $c$ of convergent sequences by ${\alpha }_1{\gamma }_{\infty }+{\sum }_{n\ge 2}{\alpha }_n{\gamma }_{n-1}$. The functional $\alpha \mapsto {\alpha }_1$ is continuous only for the first topology: $e_1-e_n\to 0$ for the second but its values remain one. Conversely $\alpha \mapsto \sum {\alpha }_n$ is continuous only for the second: $e_n\to 0$ for the first but its values remain one. More generally, comparable weak-star topologies arising from isometric preduals agree on the common closed unit ball, since its identity map is a continuous bijection from a compact space to a Hausdorff space by the Banach-Alaoglu theorem.

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.

Let $R,\alpha >0$ and ${\mathcal{C}}_{\alpha }=\{(x,t)\in {\mathbb{C}}^2:|x|<R,|t|<\alpha (1-|x|/R)\}$. The displayed norm makes the holomorphic functions with finite norm a Banach space. Finiteness forces $u(x,0)=0$: for fixed $x$, the weight grows like $1/|t|$ near zero. A Cauchy sequence in this norm converges uniformly on every compact subset of ${\mathcal{C}}_{\alpha }$, including at $t=0$. Its limit is holomorphic by locally uniform convergence of holomorphic functions, and the pointwise norm bounds show convergence in the original norm. The shrinking domain compensates for loss of an $x$-derivative in a Cauchy-Kovalevskaya theorem argument.

For positive weights $w_j$, the sequence space with norm $\Vert \alpha \Vert ={\sum }_j{w}_j|{\alpha }_j|$ is isometrically isomorphic to ${\ell }_1$: multiplication by $w_j$ is a linear bijection onto the absolutely summable sequence space, with inverse multiplication by $w_j^{-1}$. Thus it is a separable Banach space. Density can also be seen directly by truncating a weighted summable sequence and then approximating its finitely many coordinates by rational numbers.