Lemma:
Any nontrivial torus is contained in a maximal torus (so that, in particular, G contains a maximal torus).
Optional: A torus is contained in a maximal torus
Any torus is contained in a maximal torus
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We have so far proved that any compact matrix group G contains a nontrivial torus. We will now show that:
Proof:
Suppose, for a contradiction, that we have a torus T_1 not contained in a maximal torus. Then T_1 cannot itself be maximal (it's contained in itself), so it must be contained in a strictly bigger torus T_2, and T_2 is also not allowed to be maximal. Therefore T_2 is contained in a strictly bigger non-maximal torusT_3. Continuing in this manner, I can construct a sequence of nested tori T_1 inside T_2 inside T_3 etc each strictly included in the next.
We now pass to the level of Lie algebras: little t_1 inside little t_2 inside little t_3, etc. These are all subspaces of little g, which is a finite-dimensional vector space. Therefore this sequence of Lie algebras must "stabilise" at some point, that is there exists an N such that little t_{N + k} equals little t_N for all k at least 0.
We saw earlier that a torus is exp of its Lie algebra, so T_k equals exp of little t k for all k. But then T_{N + k} = T_N for all k at least 0, and this is a contradiction because T_N inside T_{N + 1} is supposed to be a strict inclusion.
Outlook
Now we have found a maximal torus t from U(1) to the n to G (with image big T), we will use it in the following way. Given a (smooth, complex) representation R from G to G L V, we can restrict to get a representation R restricted to T from T to G L V of our maximal torus. Because T is isomorphic to U(1) to the n, we deduce that V splits as a direct sum of subspaces W_{lambda} where W_lambda is the subspace of v in V such that R of t (e to the i theta_1, up to e to the i theta_n) applied to v equals e to the i (lambda_1 theta_1 plus dot dot dot plus lambda_n theta_n) times v. Here, lambda is the vector lambda_1 up to lambda_n.
Remark:
The bigger our torus, the larger n is, so the more integers lambda_i we have to split up our weight spaces.
Remark:
There is a more sophisticated way to think about lambda. We can think of lambda as the linear function lambda_1 theta_1 plus dot dot dot plus lambda_n theta_n of the thetas. The thetas are coordinates on a certain vector space. This is not quite the Lie algebra of T, because the Lie algebra of T consists of n-tuples of imaginary numbers i theta_1 up to i theta_n (just like how little u 1 is the imaginary axis). Instead, the vector theta_1 up to theta_n lives in i times little t inside the complexification of little t. This means that lambda lives most naturally in i little t, all star (where star means the dual space of linear functions). We will discuss this more formally when we talk about the Killing form.