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Transient stability constrained optimal power flow (TSOPF) is a big challenge in the field of power system operation because of its computational complexity. At present, the combination of numerical discretization method with interior point method (IPM) is considered as one of the best algorithms for large-scale TSOPF problems. However, it still suffers from the curse of dimensionality as well as unacceptable computational time and memory consumption. Although the TSOPF problem after numerical discretization is a very large nonlinear programming problem (NLP), its degrees of freedom is relatively small, which makes it very suitable for solution by a reduced-space technique. Considering this characteristic of TSOPF problems, the combination of a reduced-space technique with predictor-corrector IPM is presented in this paper as a way of relieving the computational burden of numerical discretization-based TSOPF algorithms. Several key steps of the reduced-space approach, including building basis matrices, partition strategy, computation of cross term and reduced Hessian, are discussed in detail. Detailed case studies indicate that the proposed reduced-space approach can remarkably reduce both CPU time and memory usage compared to the full-space approach, and therefore is a very promising method for solving large-scale TSOPF problems.