This paper investigates fractal dimension of linear combination of fractal continuous functions with the same or different fractal dimensions. It has been proved that: (1) BV I all fractal continuous functions with bounded variation is fractal linear space; (2) 1 D I all fractal continuous functions with Box dimension one is a fractal linear space; (3) s D I all fractal continuous functions with identical Box dimension s(1 < s ≤ 2) is surprisingly a non-fractal linear space, even non-fractal linear manifold, beyond our initial expectation, because the Box dimension of linear combination of fractal continuous functions can take any real number in [1, s) if it exists, and some different upper and lower Box dimension if it does not exit. This attracts our interests to probe into fractal characteristics of s D I , and get some suggesting results.