Give an example of a vector space V that has no proper subsp

Give an example of a vector space V that has no proper subspaces, i.e. its only subspaces are {0} and V itself

Consider set of real numbers R with usual addition \'+\' and usual multiplicaion \'*\'

Then (R, +, *) is a vector space over the field of real numbers

But R has no proper subspaces

Let \'S\' be a subspace of R and S is not equal to {0}

Now we prove that S=R

Already S is subset of R

Now we show that R is subset of S

Let a be an element of R.

Since S is not equal to {0}, there exists a non zero element say p

Now (a/p) is a scalar and p is a vector in the vector space S and

a=(a/p)p is an element of S by the definition of a vector space.

Thus every element of R can be viewed as an element of S

Hence S=R

Therefore R has no proper subspaces