There are many spaces of maps $S^1\to K$ (hence many version of the affine Grassmannian) one might want to consider. Let's list them:

• Algebraic maps (i.e. algebraic maps from $\mathbb C^\times$ to $G$ that intertwine the involution $z\mapsto \bar z^{-1}$ on $\mathbb C^\times$, and the antilinear involution on $G$ whose fixed points is $K$).

• Real-analytic maps $S^1\to K$ that extend to an analytic map $\{z\in \mathbb C: r<|z|<r^{-1}\}\to G$, for some fixed $r\in (0,1)$.

• Real-analytic maps $S^1\to K$ (the union over all $r$ of the above).

• Smooth maps $S^1\to K$.

• Continuous maps $S^1\to K$. [This version of the loop group is BAD: it does not support the basic central extension. It should never be used.]

• Sobolev-1/2 maps $S^1\to K$. Such maps are typically discontinuous, but otherwise not that bad. The group of such maps supports the basic central extension, and is in some sense maximal w.r.t. that property. [See this question of mine for more stuff about Sobolev-1/2 maps.]

The thing you call $LK$ (real analytic loops) is different from the thing you call $\mathcal{Gr}$ (algebraic loops). You can see the difference most strikingly in the case $K=S^1$ (with complexification $G=\mathbb C^\times$), where the inclusion $\mathcal{Gr}\hookrightarrow LK$ is not even dense [it's still a homotopy equivalence, though].

For $K$ semi-simple, the inclusion $\mathcal{Gr}\hookrightarrow LK$ actually is dense, but it's still not an iso (consider the maps that land in the maximal torus!) [and the inclusion is a homotopy equivalence].

There are many spaces of maps $S^1\to K$ (hence many version of the affine Grassmannian) one might want to consider. Let's list them:

• Algebraic maps (i.e. algebraic maps from $\mathbb C^\times$ to $G$ that intertwine the involution $z\mapsto \bar z^{-1}$ on $\mathbb C^\times$, and the antilinear involution on $G$ whose fixed points is $K$).

• Real-analytic maps $S^1\to K$ that extend to an analytic map $\{z\in \mathbb C: r<|z|<r^{-1}\}\to G$, for some fixed $r\in (0,1)$.

• Real-analytic maps $S^1\to K$ (the union over all $r$ of the above).

• Smooth maps $S^1\to K$.

• Continuous maps $S^1\to K$. [This version of the loop group is BAD: it does not support the basic central extension. It should never be used.]

• Sobolev-1/2 maps $S^1\to K$. Such maps are typically discontinuous, but otherwise not that bad. The group of such maps supports the basic central extension, and is in some sense maximal w.r.t. that property. [See this question of mine for more stuff about Sobolev-1/2 maps.]

The group $LK$ (real analytic loops) is different from the group $\mathcal{Gr}$ (algebraic loops). You can see the difference most strikingly in the case $K=S^1$ (with complexification $G=\mathbb C^\times$), where the inclusion $\mathcal{Gr}\hookrightarrow LK$ is not even dense [it's still a homotopy equivalence, though]. For $K$ semi-simple, the inclusion $\mathcal{Gr}\hookrightarrow LK$ actually is dense, but it's still not an iso (consider the maps that land in the maximal torus!) [and the inclusion is also a homotopy equivalence].