17.2. HAHN BANACH THEOREM 447

= Re f (x)− iRe f (ix)+Re f (y)− iRe f (iy).

Equating real parts, Re f (x+ y) = Re f (x)+Re f (y). Thus Re f is linear with respect toreal scalars as hoped.

Consider X as a real vector space and let ρ(x)≡ K||x||. Then for all x ∈M,

|Re f (x)| ≤ | f (x)| ≤ K||x||= ρ(x).

From Theorem 17.2.5, Re f may be extended to a function, h which satisfies

h(ax+by) = ah(x)+bh(y) if a,b ∈ Rh(x) ≤ K||x|| for all x ∈ X .

Actually, |h(x)| ≤ K ||x|| . The reason for this is that h(−x) = −h(x) ≤ K ||−x|| = K ||x||and therefore, h(x)≥−K ||x||. Let

F(x)≡ h(x)− ih(ix).

By arguments similar to the above, F is linear.

F (ix) = h(ix)− ih(−x)

= ih(x)+h(ix)

= i(h(x)− ih(ix)) = iF (x) .

If c is a real scalar,

F (cx) = h(cx)− ih(icx)

= ch(x)− cih(ix) = cF (x)

Now

F (x+ y) = h(x+ y)− ih(i(x+ y))

= h(x)+h(y)− ih(ix)− ih(iy)

= F (x)+F (y) .

Thus

F ((a+ ib)x) = F (ax)+F (ibx)

= aF (x)+ ibF (x)

= (a+ ib)F (x) .

This shows F is linear as claimed.Now wF(x) = |F(x)| for some |w|= 1. Therefore

|F(x)| = wF(x) = h(wx)−

must equal zero︷ ︸︸ ︷ih(iwx) = h(wx)

= |h(wx)| ≤ K||wx||= K ||x|| .

This proves the corollary.