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Linear Fisher markets are a fundamental economic model with applications in fair division as well as large-scale Internet markets. In the finite-dimensional case of $n$ buyers and $m$ items, a market equilibrium can be computed using the Eisenberg-Gale convex program. Motivated by large-scale Internet advertising and fair division applications, this paper considers a generalization of a linear Fisher market where there is a finite set of buyers and a continuum of items. We introduce generalizations of the Eisenberg-Gale convex program and its dual to this infinite-dimensional setting, which leads to Banach-space optimization problems. We establish existence of optimal solutions, strong duality, as well as necessity and sufficiency of KKT-type conditions. All these properties are established via non-standard arguments, which circumvent the limitations of duality theory in optimization over infinite-dimensional Banach spaces. Furthermore, we show that there exists a pure equilibrium allocation, i.e., a division of the item space. When the item space is a closed interval and buyers have piecewise linear valuations, we show that the Eisenberg-Gale-type convex program over the infinite-dimensional allocations can be reformulated as a finite-dimensional convex conic program, which can be solved efficiently using off-the-shelf optimization software based on primal-dual interior-point methods. Based on our convex conic reformulation, we develop the first polynomial-time cake-cutting algorithm that achieves Pareto optimality, envy-freeness, and proportionality. For general buyer valuations or a very large number of buyers, we propose computing market equilibrium using stochastic dual averaging, which finds approximate equilibrium prices with high probability. Finally, we discuss how the above results easily extend to the case of quasilinear utilities.