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Humans learn adaptively and efficiently throughout their lives. However, incrementally learning tasks causes artificial neural networks to overwrite relevant information learned about older tasks, resulting in 'Catastrophic Forgetting'. Efforts to overcome this phenomenon often utilize resources poorly, for instance, by growing the network architecture or needing to save parametric importance scores, or violate data privacy between tasks. To tackle this, we propose SPACE, an algorithm that enables a network to learn continually and efficiently by partitioning the learnt space into a Core space, that serves as the condensed knowledge base over previously learned tasks, and a Residual space, which is akin to a scratch space for learning the current task. After learning each task, the Residual is analyzed for redundancy, both within itself and with the learnt Core space. A minimal number of extra dimensions required to explain the current task are added to the Core space and the remaining Residual is freed up for learning the next task. We evaluate our algorithm on P-MNIST, CIFAR and a sequence of 8 different datasets, and achieve comparable accuracy to the state-of-the-art methods while overcoming catastrophic forgetting. Additionally, our algorithm is well suited for practical use. The partitioning algorithm analyzes all layers in one shot, ensuring scalability to deeper networks. Moreover, the analysis of dimensions translates to filter-level sparsity, and the structured nature of the resulting architecture gives us up to 5x improvement in energy efficiency during task inference over the current state-of-the-art.