sage: VectorSpace(QQ, 5).category()
Category of finite dimensional vector spaces with basis over (number fields and quotient fields and metric spaces)
sage: %time VectorSpace(QQ, 5)
CPU times: user 28 µs, sys: 9 µs, total: 37 µs
Wall time: 40.1 µs
Vector space of dimension 5 over Rational Field
sage: %time VectorSpace(QQ, 80)
CPU times: user 218 µs, sys: 1 µs, total: 219 µs
Wall time: 223 µs
Vector space of dimension 80 over Rational Field
sage: %time VectorSpace(QQ, 1000)
CPU times: user 208 µs, sys: 1 µs, total: 209 µs
Wall time: 213 µs
Vector space of dimension 1000 over Rational Field

sage: for n in 5, 80, 1000, 4000, 10000, 100000: print("n = {}".format(n)); print("Construction: ", timeit("V = VectorSpace(QQ, {})".format(n),number=1,repeat=1)); u = [ x for x i
....: n range(n) ]; v = [ x + 1 for x in range(n) ]; V = VectorSpace(QQ, n); print("Distance:     ", timeit("norm(V(u) - V(v))"))
n = 5
Construction:  1 loop, best of 1: 56.1 ms per loop
Distance:      625 loops, best of 3: 81.7 μs per loop
n = 80
Construction:  1 loop, best of 1: 159 μs per loop
Distance:      625 loops, best of 3: 299 μs per loop
n = 1000
Construction:  1 loop, best of 1: 236 μs per loop
Distance:      125 loops, best of 3: 3.18 ms per loop
n = 4000
Construction:  1 loop, best of 1: 147 μs per loop
Distance:      25 loops, best of 3: 11.3 ms per loop
n = 10000
Construction:  1 loop, best of 1: 149 μs per loop
Distance:      25 loops, best of 3: 28.7 ms per loop
n = 100000
Construction:  1 loop, best of 1: 162 μs per loop
Distance:      5 loops, best of 3: 296 ms per loop

sage: CombinatorialFreeModule(QQ, range(5)).category()
Category of finite dimensional vector spaces with basis over Rational Field
sage: %time CombinatorialFreeModule(QQ, range(5))
CPU times: user 80 µs, sys: 0 ns, total: 80 µs
Wall time: 86.1 µs
Free module generated by {0, 1, 2, 3, 4} over Rational Field
sage: %time CombinatorialFreeModule(QQ, range(80))
CPU times: user 244 µs, sys: 10 µs, total: 254 µs
Wall time: 259 µs
Free module generated by {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79} over Rational Field
sage: %time CombinatorialFreeModule(QQ, range(1000))
CPU times: user 322 µs, sys: 26 µs, total: 348 µs
Wall time: 354 µs

sage: AffineSpace(RR, 5).category()
Category of schemes over Real Field with 53 bits of precision

sage: %time AffineSpace(RR, 5)
CPU times: user 47 µs, sys: 1 µs, total: 48 µs
Wall time: 51 µs
Affine Space of dimension 5 over Real Field with 53 bits of precision
sage: %time AffineSpace(RR, 80)
CPU times: user 2 ms, sys: 39 µs, total: 2.04 ms
Wall time: 2.04 ms
Affine Space of dimension 80 over Real Field with 53 bits of precision
sage: %time AffineSpace(RR, 1000)
CPU times: user 62.8 ms, sys: 1.45 ms, total: 64.3 ms
Wall time: 63.5 ms
Affine Space of dimension 1000 over Real Field with 53 bits of precision

sage: VectorSpace(QQ, 5, inner_product_matrix=matrix.identity(5)).category()
Category of finite dimensional vector spaces with basis over (number fields and quotient fields and metric spaces)
sage: %time VectorSpace(QQ, 5, inner_product_matrix=matrix.identity(5))
CPU times: user 6.55 ms, sys: 666 µs, total: 7.22 ms
Wall time: 6.88 ms
Ambient quadratic space of dimension 5 over Rational Field
Inner product matrix:
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
sage: %time VectorSpace(QQ, 80, inner_product_matrix=matrix.identity(80))
CPU times: user 14.1 ms, sys: 439 µs, total: 14.6 ms
Wall time: 14.4 ms
Ambient quadratic space of dimension 80 over Rational Field
Inner product matrix:
sage: %time VectorSpace(QQ, 1000, inner_product_matrix=matrix.identity(1000))
CPU times: user 1.44 s, sys: 34.4 ms, total: 1.47 s
Wall time: 1.48 s
Ambient quadratic space of dimension 1000 over Rational Field
Inner product matrix: ...

sage: for n in 5, 80, 1000, 4000: print("n = {}".format(n)); print("Construction: ", timeit("V = VectorSpace(QQ, {}, inner_product_matrix=matrix.identity({}))".format(n, n),number=1,repeat=1)); u = [ x for x in range(n) ]; v = [ x + 1 for x in range(n) ]; V = VectorSpace(QQ, n, inner_product_matrix=matrix.identity(n)); print("Distance:     ", timeit("t = V(u)-V(v); sqrt(t.inner_product(t))"))
n = 5
Construction:  1 loop, best of 1: 61.9 ms per loop
Distance:      625 loops, best of 3: 49.4 μs per loop
n = 80
Construction:  1 loop, best of 1: 9.75 ms per loop
Distance:      625 loops, best of 3: 243 μs per loop
n = 1000
Construction:  1 loop, best of 1: 1.51 s per loop
Distance:      25 loops, best of 3: 14.4 ms per loop
n = 4000
Construction:  1 loop, best of 1: 23.8 s per loop
Distance:      5 loops, best of 3: 225 ms per loop

sage: EuclideanSpace(5).category()
Category of smooth manifolds over Real Field with 53 bits of precision

sage: %time EuclideanSpace(5)
CPU times: user 177 ms, sys: 10.4 ms, total: 187 ms
Wall time: 196 ms
5-dimensional Euclidean space E^5
sage: %time EuclideanSpace(80)
CPU times: user 32.8 s, sys: 758 ms, total: 33.6 s
Wall time: 31.5 s
80-dimensional Euclidean space E^80
sage: %time EuclideanSpace(1000)
(timeout)

sage: %time EuclideanSpace(5)
CPU times: user 4.87 ms, sys: 372 µs, total: 5.24 ms
Wall time: 4.93 ms
5-dimensional Euclidean space E^5
sage: %time EuclideanSpace(80)
CPU times: user 106 ms, sys: 4.52 ms, total: 110 ms
Wall time: 92 ms
80-dimensional Euclidean space E^80
sage: %time EuclideanSpace(1000)
CPU times: user 1.04 s, sys: 33.5 ms, total: 1.07 s
Wall time: 986 ms

sage: %time EuclideanSpace(1000)
CPU times: user 207 ms, sys: 5.63 ms, total: 213 ms
Wall time: 212 ms
1000-dimensional Euclidean space E^1000

sage: %time DiagonalQuadraticForm(QQ, [1]*5)
CPU times: user 60 µs, sys: 1e+03 ns, total: 61 µs
Wall time: 62.9 µs
Quadratic form in 5 variables over Rational Field with coefficients: 
[ 1 0 0 0 0 ]
[ * 1 0 0 0 ]
[ * * 1 0 0 ]
[ * * * 1 0 ]
[ * * * * 1 ]
sage: %time DiagonalQuadraticForm(QQ, [1]*80)
CPU times: user 1.35 ms, sys: 31 µs, total: 1.39 ms
Wall time: 1.44 ms
Quadratic form in 80 variables over Rational Field with coefficients: 
sage: %time DiagonalQuadraticForm(QQ, [1]*1000)
CPU times: user 144 ms, sys: 2.85 ms, total: 147 ms
Wall time: 147 ms
Quadratic form in 1000 variables over Rational Field with coefficients: