PVT PLT PENETRATED CLASS B AIRSPACE DURING DEP CLB NOT REALIZING THAT HE DID NOT HAVE CLRNC TO ENTER THAT AIRSPACE AS HE BELIEVED THAT THE TWR HAD HANDED HIM OFF TO DEP CTL.

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I WAS AT OAK INTL ARPT. I OBTAINED A VFR STANDARD DEP; WHICH INCLUDED DEST; CRUISING ALT; AND FLT ADVISORIES. I WAS ALSO ASSIGNED A XPONDER CODE. I THEN CONTACTED THE TWR; AND WAS CLRED TO TKOF ON RWY 27R; THEN TURN R TO A HDG OF 310 DEGS. OAK TWR THEN HAD ME TURN TO A HDG OF 080 DEGS AND TOLD ME TO CONTACT BAY APCH. I WAS CLBING THROUGH 2800 FT WHEN I CALLED BAY APCH AND TOLD THEM THAT I WAS CLBING THROUGH 2800 FT FOR 9500 FT. I WAS UNDER THE ASSUMPTION THAT I WAS CLRED THROUGH THE CLASS B AIRSPACE BY OAK TWR; WHO IN CONJUNCTION; I THOUGHT; WAS IN CONTACT WITH BAY APCH WHEN I WAS HANDED OFF. I WAS CLBING THROUGH 3200 FT WHEN THE BAY APCH CTLR ACKNOWLEDGED ME. I WAS TOLD TO TURN IMMEDIATELY TO A HDG OF 060 DEGS AND TO DSND TO 3000 FT; AND THAT I HAD JUST VIOLATED THE CLASS B AIRSPACE. I KNOW THAT THE CLASS B AIRSPACE IN THAT SECTOR HAS A CEILING OF 3000 FT; BUT AGAIN; I THOUGHT THAT I HAD BEEN CLRED AND IDENTED BY OAK TWR AND BAY APCH. ON THE INITIAL HDOF; I FEEL THAT THE OAK TWR CTLR SHOULD HAVE ADVISED ME TO STAY CLR OF THE CLASS B AIRSPACE; AND TO STAY AT OR BELOW 3000 FT; THEN CHK IN WITH BAY APCH AFTER OAK TWR HANDED ME OFF. I KNOW NOW NOT TO ASSUME THAT I HAVE BEEN CLRED THROUGH A CLASS B AIRSPACE; EVEN THOUGH I AM IN CONTACT WITH AIR TFC CTL UNTIL I HEAR THAT I HAVE BEEN CLRED THROUGH CLASS B AIRSPACE.