Space complexity of Coppersmith–Winograd algorithm Coppersmith–Winograd algorithm is the asymptotically fastest known algorithm for multiplying two $n \times n$ square matrices. The running time of their algorithm is $O(n^{2.376})$  which is the best known so far. What is the space complexity of this algorithm ? Is it in $\Theta(n^2)$ ?
Yes, all algorithms which stem from Strassen's original algorithm (this includes most known $n^{3-\varepsilon}$ algorithms for matrix multiplication, but not all -- see the comments) have space complexity $\Theta(n^2)$. If you could find a $n^{3-\varepsilon}$ time algorithm with $poly(\log n)$ space complexity, this would be a great advance. One application would be a $2^{(1-\varepsilon)n}$ time, $poly(n)$ space algorithm for the Subset-Sum problem. 

However there are some obstacles to such a result. For some computational models, there are fairly strong lower bounds for the time-space product of matrix multiplication. References like Yesha and Abrahamson will give you more information.